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Chapter 01

Chapter 1 introduces the concept of functions, focusing on their types, including algebraic, exponential, logarithmic, and trigonometric functions. It covers essential topics such as the slope of a line, point-slope form of linear equations, and how to model real-life situations using linear equations. The chapter also provides examples and exercises to reinforce the understanding of these concepts.

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Chapter 01

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1 Functions and Their Graphs

1000

1.1 Lines in the Plane

1.2 Functions
1.3 Graphs of Functions
V = 42.8x + 388
1.4 Shifting, Reflecting, and Stretching Graphs
0 10 1.5 Combinations of Functions
0
1.6 Inverse Functions
Section 1.7, Example 4
1.7 Linear Models and Scatter Plots
Alternative-Fueled Vehicles
Skip ODonnell/iStockphoto.com

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
2 Chapter 1 Functions and Their Graphs

Introduction to Library of Parent Functions

In Chapter 1, you will be introduced to the concept of a function. As you proceed
through the text, you will see that functions play a primary role in modeling real-life
situations.
There are three basic types of functions that have proven to be the most
important in modeling real-life situations. These functions are algebraic functions,
exponential and logarithmic functions, and trigonometric and inverse trigonometric
functions. These three types of functions are referred to as the elementary functions,
though they are often placed in the two categories of algebraic functions and
transcendental functions. Each time a new type of function is studied in detail in this
text, it will be highlighted in a box similar to those shown below. The graphs of these
functions are shown on the inside covers of this text.

ALGEBRAIC FUNCTIONS
These functions are formed by applying algebraic operations to the linear function f 共x兲 ⫽ x.
Name Function Location
Linear f 共x兲 ⫽ x Section 1.1
Quadratic f 共x兲 ⫽ x2
Section 2.1
Cubic f 共x兲 ⫽ x3 Section 2.2
1
Rational f 共x兲 ⫽ Section 2.7
x
Square root f 共x兲 ⫽ 冪x Section 1.2

TRANSCENDENTAL FUNCTIONS
These functions cannot be formed from the linear function by using algebraic operations.
Name Function Location
Exponential f 共x兲 ⫽ a , a > 0, a ⫽ 1
x
Section 3.1
Logarithmic f 共x兲 ⫽ loga x, x > 0, a > 0, a ⫽ 1 Section 3.2
Trigonometric f 共x兲 ⫽ sin x Section 4.5
f 共x兲 ⫽ cos x Section 4.5
f 共x兲 ⫽ tan x Section 4.6
f 共x兲 ⫽ csc x Section 4.6
f 共x兲 ⫽ sec x Section 4.6
f 共x兲 ⫽ cot x Section 4.6
Inverse trigonometric f 共x兲 ⫽ arcsin x Section 4.7
f 共x兲 ⫽ arccos x Section 4.7
f 共x兲 ⫽ arctan x Section 4.7

NONELEMENTARY FUNCTIONS
Some useful nonelementary functions include the following.
Name Function Location
Absolute value f 共x兲 ⫽ x ⱍⱍ Section 1.2
Greatest integer f 共x兲 ⫽ 冀x冁 Section 1.3

1.1 Lines in the Plane

The Slope of a Line What you should learn

● Find the slopes of lines.
In this section, you will study lines and their equations. The slope of a nonvertical line
● Write linear equations given
represents the number of units the line rises or falls vertically for each unit of
points on lines and their slopes.
horizontal change from left to right. For instance, consider the two points
● Use slope-intercept forms of
共x1, y1兲 and 共x2, y2 兲 linear equations to sketch lines.
● Use slope to identify parallel and
on the line shown in Figure 1.1.
perpendicular lines.
y Why you should learn it
The slope of a line can be used to
(x2 , y2) solve real-life problems. For instance,
y2 in Exercise 97 on page 14, you will
y 2 − y1 use a linear equation to model
(x1 , y1) student enrollment at Penn State
y1
University.
x 2 − x1

x
x1 x2

Figure 1.1

As you move from left to right along this line, a change of

共 y2 ⫺ y1兲 units in the vertical direction corresponds to a change
of 共x2 ⫺ x1兲 units in the horizontal direction. That is,
y2 ⫺ y1 ⫽ the change in y
and
x2 ⫺ x1 ⫽ the change in x.
The slope of the line is given by the ratio of these two changes.

Definition of the Slope of a Line

The slope m of the nonvertical line through 共x1, y1兲 and 共x2, y2 兲 is
y2 ⫺ y1 change in y
m⫽ ⫽
x2 ⫺ x1 change in x
where x1 ⫽ x 2.

When this formula for slope is used, the order of subtraction is important. Given
two points on a line, you are free to label either one of them as 共x1, y1兲 and the other
as 共x2, y2 兲. Once you have done this, however, you must form the numerator and
denominator using the same order of subtraction.
y2 ⫺ y1 y1 ⫺ y2 y2 ⫺ y1
m⫽ m⫽ m⫽
x2 ⫺ x1 x1 ⫺ x2 x1 ⫺ x2

Correct Correct Incorrect

Throughout this text, the term line always means a straight line.
Kurhan 2010/used under license from Shutterstock.com

Example 1 Finding the Slope of a Line

Find the slope of the line passing through each pair of points. Explore the Concept
a. 共⫺2, 0兲 and 共3, 1兲 Use a graphing utility
b. 共⫺1, 2兲 and 共2, 2兲 to compare the slopes
c. 共0, 4兲 and 共1, ⫺1兲 of the lines y ⫽ 0.5x,
y ⫽ x, y ⫽ 2x, and y ⫽ 4x.
Solution What do you observe about
Difference in y-values these lines? Compare the slopes
of the lines y ⫽ ⫺0.5x,
y2 ⫺ y1 1⫺0 1 1 y ⫽ ⫺x, y ⫽ ⫺2x, and
a. m ⫽ ⫽ ⫽ ⫽ y ⫽ ⫺4x. What do you observe
x2 ⫺ x1 3 ⫺ 共⫺2兲 3 ⫹ 2 5
about these lines? (Hint: Use a
Difference in x-values square setting to obtain a true
geometric perspective.)
2⫺2 0
b. m ⫽ ⫽ ⫽0
2 ⫺ 共⫺1兲 3
⫺1 ⫺ 4 ⫺5
c. m ⫽ ⫽ ⫽ ⫺5
1⫺0 1
The graphs of the three lines are shown in Figure 1.2. Note that the square setting gives
the correct “steepness” of the lines.

4 4 6

(−1, 2) (2, 2) (0, 4)

(3, 1)
−4 5 −4 5
−4 8
(−2, 0) (1, − 1)
−2 −2 −2
(a) (b) (c)
Figure 1.2

Now try Exercise 15.

The definition of slope does not apply to vertical lines. 5

For instance, consider the points 共3, 4兲 and 共3, 1兲 on the (3, 4)
vertical line shown in Figure 1.3. Applying the formula
for slope, you obtain (3, 1)
4⫺1 3 −1 8
m⫽ ⫽ . Undefined
3⫺3 0 −1

Because division by zero is undefined, the slope of a Figure 1.3

vertical line is undefined.
From the slopes of the lines shown in Figures 1.2 and 1.3,
you can make the following generalizations about the slope of a line.

The Slope of a Line

1. A line with positive slope 共m > 0兲 rises from left to right.
2. A line with negative slope 共m < 0兲 falls from left to right.
3. A line with zero slope 共m ⫽ 0兲 is horizontal.
4. A line with undefined slope is vertical.

The Point-Slope Form of the Equation of a Line

y
When you know the slope of a line and you also
know the coordinates of one point on the line, (x , y )
you can find an equation of the line. For
instance, in Figure 1.4, let 共x1, y1兲 be a point y − y1
(x1, y1)
on the line whose slope is m. When 共x, y兲 is any
other point on the line, it follows that x − x1
y ⫺ y1
⫽ m.
x ⫺ x1
x
This equation in the variables x and y can
be rewritten in the point-slope form of the
Figure 1.4
equation of a line.

Point-Slope Form of the Equation of a Line

The point-slope form of the equation of the line that passes through the point
共x1, y1兲 and has a slope of m is
y ⫺ y1 ⫽ m共x ⫺ x1兲.

Example 2 The Point-Slope Form of the Equation of a Line

Find an equation of the line that passes through the point
共1, ⫺2兲
and has a slope of 3.

Solution y = 3x − 5
3

y ⫺ y1 ⫽ m共x ⫺ x1兲 Point-slope form

−5 10
y ⫺ 共⫺2兲 ⫽ 3共x ⫺ 1兲 Substitute for y1, m, and x1.
(1, − 2)
y ⫹ 2 ⫽ 3x ⫺ 3 Simplify.

y ⫽ 3x ⫺ 5 Solve for y. −7

The line is shown in Figure 1.5. Figure 1.5

Now try Exercise 25.

The point-slope form can be used to find an equation of a nonvertical line passing
through two points
Study Tip
共x1, y1兲 and 共x2, y2 兲.
When you find an
First, find the slope of the line. equation of the line
y2 ⫺ y1 that passes through
m⫽ , x1 ⫽ x2 two given points, you need to
x2 ⫺ x1
substitute the coordinates of
only one of the points into the
Then use the point-slope form to obtain the equation point-slope form. It does not
y2 ⫺ y1 matter which point you choose
y ⫺ y1 ⫽ 共x ⫺ x1兲.
x2 ⫺ x1 because both points will yield
the same result.
This is sometimes called the two-point form of the equation of a line.

Example 3 A Linear Model for Profits Prediction

During 2006, Research In Motion’s net profits were $631.6 million, and in 2007 net
profits were $1293.9 million. Write a linear equation giving the net profits y in terms of
the year x. Then use the equation to predict the net profits for 2008. (Source: Research
In Motion Limited)
y
Solution

Net profits (in millions of dollars)

Let x ⫽ 0 represent 2000. In Figure 1.6, let 3500 y = 662.3x − 3342.2
共6, 631.6兲 and 共7, 1293.9兲 be two points on 3000
the line representing the net profits. The slope
of this line is 2500

2000
1293.9 ⫺ 631.6 (8, 1956.2)
m⫽ ⫽ 662.3.
7⫺6 1500
(7, 1293.9)
By the point-slope form, the equation of the 1000
line is as follows. 500 (6, 631.6)
y ⫺ 631.6 ⫽ 662.3共x ⫺ 6兲 x
6 7 8 9 10
y ⫽ 662.3x ⫺ 3342.2 Year (6 ↔ 2006)
Now, using this equation, you can predict the Figure 1.6
2008 net profits 共x ⫽ 8兲 to be
y ⫽ 662.3共8兲 ⫺ 3342.2 ⫽ 5298.4 ⫺ 3342.2 ⫽ $1956.2 million.
(In this case, the prediction is quite good––the actual net profits in 2008
were $1968.8 million.)
Now try Exercise 33.

Library of Parent Functions: Linear Function

In the next section, you will be introduced to the precise meaning of the term
function. The simplest type of function is the parent linear function
f 共x兲 ⫽ x.
As its name implies, the graph of the parent linear function is a line. The basic
characteristics of the parent linear function are summarized below and on the inside
cover of this text. (Note that some of the terms below will be defined later in the
text.)

Graph of f 共x兲 ⫽ x y

Domain: 共⫺ ⬁, ⬁兲
Range: 共⫺ ⬁, ⬁兲
Elliot Westacott 2010/used under license from Shutterstock.com

f(x) = x
Intercept: 共0, 0兲 x
(0, 0)
Increasing

The function f 共x兲 ⫽ x is also referred to as the identity function. Later in this text,
you will learn that the graph of the linear function f 共x兲 ⫽ mx ⫹ b is a line with
slope m and y-intercept 共0, b兲. When m ⫽ 0, f 共x兲 ⫽ b is called a constant function
and its graph is a horizontal line.

Sketching Graphs of Lines

Many problems in coordinate geometry can be classified as follows.
1. Given a graph (or parts of it), find its equation.
2. Given an equation, sketch its graph.
For lines, the first problem is solved easily by using the point-slope form.
This formula, however, is not particularly useful for solving the second type of
problem. The form that is better suited to graphing linear equations is the slope-
intercept form of the equation of a line, y ⫽ mx ⫹ b.

Slope-Intercept Form of the Equation of a Line

The graph of the equation
y ⫽ mx ⫹ b
is a line whose slope is m and whose y-intercept is 共0, b兲.

Example 4 Using the Slope-Intercept Form

Determine the slope and y-intercept of each linear equation. Then describe its graph.
a. x ⫹ y ⫽ 2
b. y ⫽ 2

Algebraic Solution Graphical Solution

a. Begin by writing the equation in slope-intercept form. a.
x⫹y⫽2 Write original equation.

y⫽2⫺x Subtract x from each side.

y ⫽ ⫺x ⫹ 2 Write in slope-intercept form.

From the slope-intercept form of the equation, the slope

is ⫺1 and the y-intercept is
共0, 2兲.
Because the slope is negative, you know that the graph of
the equation is a line that falls one unit for every unit it
moves to the right.
b. By writing the equation y ⫽ 2 in slope-intercept form
b.
y ⫽ 共0兲x ⫹ 2
you can see that the slope is 0 and the y-intercept is
共0, 2兲.
A zero slope implies that the line is horizontal.

Now try Exercise 35.

From the slope-intercept form of the equation of a line, you can see that a
horizontal line 共m ⫽ 0兲 has an equation of the form
y ⫽ b. Horizontal line

This is consistent with the fact that each point on a horizontal line through 共0, b兲 has
a y-coordinate of b. Similarly, each point on a vertical line through 共a, 0兲 has an
x-coordinate of a. So, a vertical line has an equation of the form
x ⫽ a. Vertical line

This equation cannot be written in slope-intercept form because the slope of a vertical
line is undefined. However, every line has an equation that can be written in the
general form
Ax ⫹ By ⫹ C ⫽ 0 General form of the equation of a line

where A and B are not both zero.

Summary of Equations of Lines

1. General form: Ax ⫹ By ⫹ C ⫽ 0
2. Vertical line: x⫽a
3. Horizontal line: y⫽b
4. Slope-intercept form: y ⫽ mx ⫹ b
5. Point-slope form: y ⫺ y1 ⫽ m共x ⫺ x1兲

Example 5 Different Viewing Windows

When a graphing utility is used to graph a line, it is important to realize that the graph
of the line may not visually appear to have the slope indicated by its equation. This
occurs because of the viewing window used for the graph. For instance, Figure 1.7
shows graphs of y ⫽ 2x ⫹ 1 produced on a graphing utility using three different viewing
windows. Notice that the slopes in Figures 1.7(a) and (b) do not visually appear to be
equal to 2. When you use a square setting, as in Figure 1.7(c), the slope visually appears
to be 2.

10 20
Using a nonsquare setting, you y = 2x + 1
y = 2x + 1 do not obtain a graph with a true
− 10 10 geometric perspective. So, the −3 3
slope does not visually appear
to be 2.
− 10 −20

(a) Nonsquare setting (b) Nonsquare setting

y = 2x + 1 Using a square setting, you

− 15 15 can obtain a graph with a true
geometric perspective. So, the
slope visually appears to be 2.
− 10

(c) Square setting

Figure 1.7

Now try Exercise 61.

Parallel and Perpendicular Lines

The slope of a line is a convenient tool for determining whether two lines are parallel Explore the Concept
or perpendicular.
Graph the lines
y1 ⫽ 12 x ⫹ 1 and
Parallel Lines y2 ⫽ ⫺2x ⫹ 1 in the
Two distinct nonvertical lines are parallel if and only if their slopes are equal. same viewing window. What do
That is, you observe?

m1 ⫽ m2. Graph the lines y1 ⫽ 2x ⫹ 1,

y2 ⫽ 2x, and y3 ⫽ 2x ⫺ 1 in
the same viewing window.
What do you observe?
Example 6 Equations of Parallel Lines
Find the slope-intercept form of the equation of the line that passes through the point
共2, ⫺1兲 and is parallel to the line
2x ⫺ 3y ⫽ 5.

Solution
Begin by writing the equation of the given line in slope-intercept form.
2x ⫺ 3y ⫽ 5 Write original equation.

⫺2x ⫹ 3y ⫽ ⫺5 Multiply by ⫺1.

3y ⫽ 2x ⫺ 5 Add 2x to each side.

2 5
y⫽ x⫺ Write in slope-intercept form.
3 3
Therefore, the given line has a slope of
m ⫽ 23.
2
Any line parallel to the given line must also have a slope of 3. So, the line through
共2, ⫺1兲 has the following equation.
y ⫺ y1 ⫽ m共x ⫺ x1兲 Point-slope form

2
y ⫺ 共⫺1兲 ⫽ 共x ⫺ 2兲 Substitute for y1, m, and x1.
3
2 4
y⫹1⫽ x⫺ Simplify. y = 23 x − 5
3 3 1 3

2 7
y⫽ x⫺ Write in slope-intercept form. −1 5
3 3
(2, − 1)
Notice the similarity between the slope-intercept form of the original equation and the
slope-intercept form of the parallel equation. The graphs of both equations are shown
in Figure 1.8. −3 y = 23 x − 7
3

Now try Exercise 67(a). Figure 1.8

Perpendicular Lines
Two nonvertical lines are perpendicular if and only if their slopes are negative
reciprocals of each other. That is,
1
m1 ⫽ ⫺ .
m2

Example 7 Equations of Perpendicular Lines

Find the slope-intercept form of the equation of the line that passes through the point
共2, ⫺1兲 and is perpendicular to the line
2x ⫺ 3y ⫽ 5.

Solution
From Example 6, you know that the equation can be written in the slope-intercept form
y ⫽ 23 x ⫺ 53.

You can see that the line has a slope of 32. So, any line perpendicular to this line must
have a slope of ⫺ 2 共because ⫺ 2 is the negative reciprocal of 3 兲. So, the line through the
3 3 2

point 共2, ⫺1兲 has the following equation.

y ⫺ 共⫺1兲 ⫽ ⫺ 32共x ⫺ 2兲 Write in point-slope form.

y⫹1⫽ ⫺ 32x ⫹3 Simplify.

y⫽ ⫺ 32x ⫹2 Write in slope-intercept form.

The graphs of both equations are shown in Figure 1.9.

y = 23 x − 5
3
3

−2 7
(2, − 1) What’s Wrong?
You use a graphing utility
−3
y = − 32 x + 2 to graph y1 ⫽ 1.5x and
y2 ⫽ ⫺1.5x ⫹ 5, as shown in
Figure 1.9 the figure. You use the graph
Now try Exercise 67(b). to conclude that the lines are
perpendicular. What’s wrong?

Example 8 Graphs of Perpendicular Lines 10

Use a graphing utility to graph the lines y ⫽ x ⫹ 1 and y ⫽ ⫺x ⫹ 3 in the same

viewing window. The lines are supposed to be perpendicular (they have slopes of −10 10
m1 ⫽ 1 and m2 ⫽ ⫺1). Do they appear to be perpendicular on the display?

Solution −10
When the viewing window is nonsquare, as in Figure 1.10, the two lines will not appear
perpendicular. When, however, the viewing window is square, as in Figure 1.11, the
lines will appear perpendicular.

y = −x + 3 10
y=x +1 y = −x + 3 10
y =x +1

− 10 10 − 15 15

− 10 −10

Figure 1.10 Figure 1.11

Now try Exercise 81.

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1.1 Exercises For instructions on how to use a graphing utility, see Appendix A.

Vocabulary and Concept Check

1. Match each equation with its form.
(a) Ax ⫹ By ⫹ C ⫽ 0 (i) vertical line
(b) x ⫽ a (ii) slope-intercept form
(c) y ⫽ b (iii) general form
(d) y ⫽ mx ⫹ b (iv) point-slope form
(e) y ⫺ y1 ⫽ m共x ⫺ x 1兲 (v) horizontal line

In Exercises 2 and 3, fill in the blank.

2. For a line, the ratio of the change in y to the change in x is called the _______ of the line.
3. Two lines are _______ if and only if their slopes are equal.
1
4. What is the relationship between two lines whose slopes are ⫺3 and ?
3
5. What is the slope of a line that is perpendicular to the line represented by x ⫽ 3?
1
6. Give the coordinates of a point on the line whose equation in point-slope form is y ⫺ 共⫺2兲 ⫽ 共x ⫺ 5兲.
2
Procedures and Problem Solving
Using Slope In Exercises 7 and 8, identify the line that Sketching Lines In Exercises 11 and 12, sketch the lines
has the indicated slope. through the point with the indicated slopes on the same
2 set of coordinate axes.
7. (a) m ⫽ 3 (b) m is undefined. (c) m ⫽ ⫺2
y Point Slopes
11. 共2, 3兲 (a) 0 (b) 1 (c) 2 (d) ⫺3
12. 共⫺4, 1兲
1
L2 (a) 3 (b) ⫺3 (c) 2 (d) Undefined
x
Finding the Slope of a Line In Exercises 13–16, find the
L1 slope of the line passing through the pair of points. Then
L3
use a graphing utility to plot the points and use the draw
3
feature to graph the line segment connecting the two
8. (a) m ⫽ 0 (b) m ⫽ ⫺ 4 (c) m ⫽ 1 points. (Use a square setting.)
y
13. 共0, ⫺10兲, 共⫺4, 0兲 14. 共2, 4兲, 共4, ⫺4兲
L1
L3 15. 共⫺6, ⫺1兲, 共⫺6, 4兲 16. 共⫺3, ⫺2兲, 共1, 6兲

x Using Slope In Exercises 17–24, use the point on the

L2
line and the slope of the line to find three additional
points through which the line passes. (There are many
correct answers.)
Point Slope
Estimating Slope In Exercises 9 and 10, estimate the
slope of the line. 17. 共2, 1兲 m⫽0

9. y 10. y 18. 共3, ⫺2兲 m⫽0

8 8 19. 共1, 5兲 m is undefined.
6 6 20. 共⫺4, 1兲 m is undefined.
4 4
21. 共0, ⫺9兲 m ⫽ ⫺2
2 2
x x
22. 共⫺5, 4兲 m⫽2
2 4 6 8 4 6 8 23. 共7, ⫺2兲 m ⫽ 12
24. 共⫺1, ⫺6兲 m ⫽ ⫺2
1

The Point-Slope Form of the Equation of a Line In Finding the Slope-Intercept Form In Exercises 51–60,
Exercises 25–32, find an equation of the line that passes write an equation of the line that passes through the
through the given point and has the indicated slope. points. Use the slope-intercept form (if possible). If not
Sketch the line by hand. Use a graphing utility to verify possible, explain why and use the general form. Use a
your sketch, if possible. graphing utility to graph the line (if possible).
25. 共0, ⫺2兲, m ⫽ 3 26. 共⫺3, 6兲, m ⫽ ⫺2 51. 共5, ⫺1兲, 共⫺5, 5兲 52. 共4, 3兲, 共⫺4, ⫺4兲
27. 共2, ⫺3兲, m ⫽ ⫺ 2 1
28. 共⫺2, ⫺5兲, m ⫽ 34 53. 共⫺8, 1兲, 共⫺8, 7兲 54. 共⫺1, 4兲, 共6, 4兲
29. 共6, ⫺1兲, m is undefined 55. 共2, 12 兲, 共12, 54 兲 56. 共1, 1兲, 共6, ⫺ 23 兲
30. 共⫺10, 4兲, m is undefined 57. 共⫺ 101 , ⫺ 35 兲, 共109 , ⫺ 95 兲 58. 共34, 32 兲, 共⫺ 43, 74 兲
31. 共⫺ 12, 32 兲, m ⫽ 0 32. 共2.3, ⫺8.5兲, m ⫽ 0 59. 共1, 0.6兲, 共⫺2, ⫺0.6兲 60. 共⫺8, 0.6兲, 共2, ⫺2.4兲
33. Finance The median player salary for the New York Different Viewing Windows In Exercises 61 and 62, use
Yankees was $1.6 million in 2001 and $5.2 million in a graphing utility to graph the equation using each viewing
2009. Write a linear equation giving the median salary y window. Describe the differences in the graphs.
in terms of the year x. Then use the equation to predict the
median salary in 2017. 61. y ⫽ 0.5x ⫺ 3
34. Finance The median player salary for the Dallas Xmin = -5 Xmin = -2 Xmin = -5
Cowboys was $441,300 in 2000 and $1,326,720 in Xmax = 10 Xmax = 10 Xmax = 10
2008. Write a linear equation giving the median salary y Xscl = 1 Xscl = 1 Xscl = 1
in terms of the year x. Then use the equation to predict the Ymin = -1 Ymin = -4 Ymin = -7
median salary in 2016. Ymax = 10 Ymax = 1 Ymax = 3
Yscl = 1 Yscl = 1 Yscl = 1
Using the Slope-Intercept Form In Exercises 35–42,
determine the slope and y-intercept (if possible) of the
linear equation. Then describe its graph. 62. y ⫽ ⫺8x ⫹ 5

35. x ⫺ 2y ⫽ 4 36. 3x ⫹ 4y ⫽ 1 Xmin = -5 Xmin = -5 Xmin = -5

37. 2x ⫺ 5y ⫹ 10 ⫽ 0 38. 4x ⫺ 3y ⫺ 9 ⫽ 0 Xmax = 5 Xmax = 10 Xmax = 13
Xscl = 1 Xscl = 1 Xscl = 1
39. x ⫽ ⫺6 40. y ⫽ 12
Ymin = -10 Ymin = -80 Ymin = -2
41. 3y ⫹ 2 ⫽ 0 42. 2x ⫺ 5 ⫽ 0 Ymax = 10 Ymax = 80 Ymax = 10
Yscl = 1 Yscl = 20 Yscl = 1
Using the Slope-Intercept Form In Exercises 43–48,
(a) find the slope and y-intercept (if possible) of the
equation of the line algebraically, and (b) sketch the line Parallel and Perpendicular Lines In Exercises 63–66,
by hand. Use a graphing utility to verify your answers to determine whether the lines L1 and L2 passing through
parts (a) and (b). the pairs of points are parallel, perpendicular, or neither.
43. 5x ⫺ y ⫹ 3 ⫽ 0 44. 2x ⫹ 3y ⫺ 9 ⫽ 0 63. L1: 共0, ⫺1兲, 共5, 9兲 64. L1: 共⫺2, ⫺1兲, 共1, 5兲
45. 5x ⫺ 2 ⫽ 0 46. 3x ⫹ 7 ⫽ 0 L2: 共0, 3兲, 共4, 1兲 L2: 共1, 3兲, 共5, ⫺5兲
47. 3y ⫹ 5 ⫽ 0 48. ⫺11 ⫺ 8y ⫽ 0 65. L1: 共3, 6兲, 共⫺6, 0兲 66. L1: (4, 8), (⫺4, 2)
L2: 共0, ⫺1兲, 共5, 73 兲 L2: 共3, ⫺5兲, 共⫺1, 13 兲
Finding the Slope-Intercept Form In Exercises 49 and
50, find the slope-intercept form of the equation of the Equations of Parallel and Perpendicular Lines In
line shown. Exercises 67–76, write the slope-intercept forms of the
49. y 50. y equations of the lines through the given point (a) parallel
to the given line and (b) perpendicular to the given line.
2
)− 1, 32 )
x 67. 共2, 1兲, 4x ⫺ 2y ⫽ 3 68. 共⫺3, 2兲, x⫹y⫽7
−4 −2
共⫺ 3, 8 兲, 3x ⫹ 4y ⫽ 7 70. 共25, ⫺1兲,
4 2 7
−2 69. 3x ⫺ 2y ⫽ 6
(1, − 3) x
−4 −2 2 71. 共⫺3.9, ⫺1.4兲, 6x ⫹ 2y ⫽ 9
−2 (4, − 1)
(−1, − 7) 72. 共⫺1.2, 2.4兲, 5x ⫹ 4y ⫽ 1
−4 73. 共3, ⫺2兲, x ⫺ 4 ⫽ 0 74. 共3, ⫺1兲, y⫺2⫽0
75. 共⫺4, 1兲, y ⫹ 2 ⫽ 0 76. 共⫺2, 4兲, x⫹5⫽0

Equations of Parallel Lines In Exercises 77 and 78, the 87. MODELING DATA
lines are parallel. Find the slope-intercept form of the
The graph shows the sales y (in millions of dollars) of
equation of line y2 .
the Coca-Cola Bottling Company each year x from 2000
77. y 78. y through 2008, where x ⫽ 0 represents 2000. (Source:
5 4 Coca-Cola Bottling Company)
y2
y1 = 2x + 4 y
y2 y1 = −2x + 1

Sales (in millions of dollars)

(− 1, 1) 1500 (8, 1464)
1 x (6, 1431)
x −3 −2 −1 2 3 4 1400 (7, 1436)
−4 −3 1 2 3 −2 1300 (2, 1247) (5, 1380)
(−1, − 1) −3 (4, 1257)
−3 −4 1200
(3, 1211)
1100
(0, 995)
Equations of Perpendicular Lines In Exercises 79 and 1000 (1, 1023)
80, the lines are perpendicular. Find the slope-intercept 900
form of the equation of line y2 . x
0 1 2 3 4 5 6 7 8
79. y 80. y
Year (0 ↔ 2000)
5 6
4 (a) Use the slopes to determine the years in which the sales
3 (− 3, 5) y2
2
showed the greatest increase and greatest decrease.
y1 = 2x + 3
(−2, 2) x (b) Find the equation of the line between the years
x −4 −2 2 4 6
2000 and 2008.
−3 −1 1 2
y2 −4
−2 y1 = 3x − 4 (c) Interpret the meaning of the slope of the line from
−3 part (b) in the context of the problem.
(d) Use the equation from part (b) to estimate the sales
Graphs of Parallel and Perpendicular Lines In Exercises of the Coca-Cola Bottling Company in 2010. Do
81–84, identify any relationships that exist among the you think this is an accurate estimate? Explain.
lines, and then use a graphing utility to graph the three
equations in the same viewing window. Adjust the viewing
window so that each slope appears visually correct. Use 88. MODELING DATA
the slopes of the lines to verify your results. The table shows the profits y
1
81. (a) y ⫽ 2x (b) y ⫽ ⫺2x (c) y ⫽ 2x
(in millions of dollars) for
Buffalo Wild Wings for each
82. (a) y ⫽ 23x (b) y ⫽ ⫺ 32x (c) y ⫽ ⫹ 2 2
3x
year x from 2002 through 2008,
83. (a) y ⫽ ⫺ 12x (b) y ⫽ ⫺ 12x ⫹ 3 (c) y ⫽ 2x ⫺ 4 where x ⫽ 2 represents 2002.
84. (a) y⫽x⫺8 (b) y⫽x⫹1 (c) y ⫽ ⫺x ⫹ 3 (Source: Buffalo Wild Wings Inc.)

85. Architectural Design The “rise to run” ratio of the (a) Sketch a
Year, x graph of Profits, y
roof of a house determines the steepness of the roof.
The rise to run ratio of the roof in the figure is 3 to 4. 2 3.1 the data.
Determine the maximum height in the attic of the house 3 3.9 (b) Use the slopes
if the house is 32 feet wide. 4 7.2 to determine the
attic height 5 8.9 years in which the
4 6 16.3 profits showed the
3 greatest and least
7 19.7
increases.
8 24.4
(c) Find the equation of the
line between the years
32 ft 2002 and 2008.
86. Highway Engineering When driving down a mountain (d) Interpret the meaning of the slope of the line from
road, you notice warning signs indicating that it is a part (c) in the context of the problem.
“12% grade.” This means that the slope of the road (e) Use the equation from part (c) to estimate the profit
12
is ⫺ 100. Approximate the amount of horizontal change for Buffalo Wild Wings in 2010. Do you think this
in your position if you note from elevation markers that is an accurate estimate? Explain.
you have descended 2000 feet vertically.
Sean Locke/iStockphoto.com

Using a Rate of Change to Write an Equation In Exercises 96. Real Estate A real estate office handles an apartment
89–92, you are given the dollar value of a product in 2009 complex with 50 units. When the rent per unit is $580
and the rate at which the value of the product is expected per month, all 50 units are occupied. However, when
to change during the next 5 years. Write a linear equation the rent is $625 per month, the average number of
that gives the dollar value V of the product in terms of occupied units drops to 47. Assume that the relationship
the year t. (Let t ⴝ 9 represent 2009.) between the monthly rent p and the demand x is linear.
2009 Value Rate (a) Write the equation of the line giving the demand x
in terms of the rent p.
89. $2540 $125 increase per year
(b) Use a graphing utility to graph the demand equation
90. $156 $4.50 increase per year
and use the trace feature to estimate the number
91. $20,400 $2000 decrease per year of units occupied when the rent is $655. Verify
92. $245,000 $5600 decrease per year your answer algebraically.
(c) Use the demand equation to predict the number of
93. Accounting A school district purchases a high-
units occupied when the rent is lowered to $595.
volume printer, copier, and scanner for $25,000. After
Verify your answer graphically.
10 years, the equipment will have to be replaced. Its
value at that time is expected to be $2000. 97. (p. 3) In 1990, Penn State
University had an enrollment of 75,365
(a) Write a linear equation giving the value V of the
students. By 2009, the enrollment had
equipment for each year t during its 10 years of use.
increased to 87,163. (Source: Penn State
(b) Use a graphing utility to graph the linear equation Fact Book)
representing the depreciation of the equipment, and
(a) What was the average annual change in
use the value or trace feature to complete the table.
enrollment from 1990 to 2009?
Verify your answers algebraically by using the
equation you found in part (a). (b) Use the average annual change in enrollment to
estimate the enrollments in 1995, 2000, and 2005.
t 0 1 2 3 4 5 6 7 8 9 10 (c) Write the equation of a line that represents the
given data. What is its slope? Interpret the slope in
V
the context of the problem.
98. Writing Using the results of Exercise 97, write a
94. Meterology Recall that water freezes at 0⬚C 共32⬚F兲
short paragraph discussing the concepts of slope and
and boils at 100⬚C 共212⬚F兲.
average rate of change.
(a) Find an equation of the line that shows the relationship
between the temperature in degrees Celsius C and Conclusions
degrees Fahrenheit F.
True or False? In Exercises 99 and 100, determine
(b) Use the result of part (a) to complete the table.
whether the statement is true or false. Justify your
answer.
C ⫺10⬚ 10⬚ 177⬚
99. The line through 共⫺8, 2兲 and 共⫺1, 4兲 and the line
F 0⬚ 68⬚ 90⬚ through 共0, ⫺4兲 and 共⫺7, 7兲 are parallel.
95. Business A contractor purchases a bulldozer for 100. If the points 共10, ⫺3兲 and 共2, ⫺9兲 lie on the same line,
$36,500. The bulldozer requires an average expenditure then the point 共⫺12, ⫺ 372 兲 also lies on that line.
of $9.25 per hour for fuel and maintenance, and the
operator is paid $18.50 per hour. Exploration In Exercises 101–104, use a graphing utility
to graph the equation of the line in the form
(a) Write a linear equation giving the total cost C of
operating the bulldozer for t hours. (Include the x y
ⴙ ⴝ 1, a ⴝ 0, b ⴝ 0.
purchase cost of the bulldozer.) a b
(b) Assuming that customers are charged $65 per hour Use the graphs to make a conjecture about what a and b
of bulldozer use, write an equation for the revenue represent. Verify your conjecture.
R derived from t hours of use.
x y x y
(c) Use the profit formula 共P ⫽ R ⫺ C兲 to write an 101. ⫹ ⫽1 102. ⫹ ⫽1
equation for the profit gained from t hours of use. 5 ⫺3 ⫺6 2
x y x y
(d) Use the result of part (c) to find the break-even 103. ⫹ 2⫽1 104. 1 ⫹ ⫽1
point (the number of hours the bulldozer must be 4 ⫺3 2 5
used to gain a profit of 0 dollars).
Kurhan 2010/used under license from Shutterstock.com

Using Intercepts In Exercises 105–108, use the results 115. Think About It Does every line have an infinite
of Exercises 101–104 to write an equation of the line that number of lines that are parallel to it? Explain.
passes through the points.
105. x-intercept: 共2, 0兲 106. x-intercept: 共⫺5, 0兲 116. C A P S T O N E Match the description with its graph.
Determine the slope of each graph and how it is
y-intercept: 共0, 3兲 y-intercept: 共0, ⫺4兲 interpreted in the given context. [The graphs are
107. x-intercept: 共⫺ 16, 0兲 108. x-intercept: 共34, 0兲 labeled (i), (ii), (iii), and (iv).]
y-intercept: 共0, ⫺ 23 兲 y-intercept: 共0, 45 兲 (i) 40 (ii) 125

Think About It In Exercises 109 and 110, determine

which equation(s) may be represented by the graphs
shown. (There may be more than one correct answer.)
0 8 0 10
y y 0 0
109. 110.
(iii) 25 (iv) 600

x
0 10 0 6
0 0
x
(a) You are paying $10 per week to repay a $100 loan.
(a) 2x ⫺ y ⫽ ⫺10 (a) 2x ⫹ y ⫽ 5 (b) An employee is paid $12.50 per hour plus $1.50
(b) 2x ⫹ y ⫽ 10 (b) 2x ⫹ y ⫽ ⫺5 for each unit produced per hour.
(c) x ⫺ 2y ⫽ 10 (c) x ⫺ 2y ⫽ 5 (c) A sales representative receives $30 per day for
food plus $0.35 for each mile traveled.
(d) x ⫹ 2y ⫽ 10 (d) x ⫺ 2y ⫽ ⫺5
(d) A computer that was purchased for $600 depreciates
Think About It In Exercises 111 and 112, determine $100 per year.
which pair of equations may be represented by the
graphs shown.
Cumulative Mixed Review
111. y 112. y
Identifying Polynomials In Exercises 117–122, determine
whether the expression is a polynomial. If it is, write the
polynomial in standard form.
x 117. x ⫹ 20 118. 3x ⫺ 10x2 ⫹ 1
x
119. 4x2 ⫹ x⫺1 ⫺ 3 120. 2x2 ⫺ 2x4 ⫺ x3 ⫹ 2
x2 ⫹ 3x ⫹ 4
121. 122. 冪x2 ⫹ 7x ⫹ 6
x2 ⫺ 9
(a) 2x ⫺ y ⫽ 5 (a) 2x ⫺ y ⫽ 2
2x ⫺ y ⫽ 1 x ⫹ 2y ⫽ 12 Factoring Trinomials In Exercises 123–126, factor the
trinomial.
(b) 2x ⫹ y ⫽ ⫺5 (b) x ⫺ y ⫽ 1
2x ⫹ y ⫽ 1 x⫹y⫽6 123. x2 ⫺ 6x ⫺ 27 124. x2 ⫺ 11x ⫹ 28
(c) 2x ⫺ y ⫽ ⫺5 (c) 2x ⫹ y ⫽ 2 125. 2x2 ⫹ 11x ⫺ 40 126. 3x2 ⫺ 16x ⫹ 5
2x ⫺ y ⫽ 1 x ⫺ 2y ⫽ 12
127. Make a Decision To work an extended application
(d) x ⫺ 2y ⫽ ⫺5 (d) x ⫺ 2y ⫽ 2
analyzing the numbers of bachelor’s degrees earned by
x ⫺ 2y ⫽ ⫺1 x ⫹ 2y ⫽ 12 women in the United States from 1985 through 2007,
visit this textbook’s Companion Website. (Source:
113. Think About It Does every line have both an x-intercept
National Center for Education Statistics)
and a y-intercept? Explain.
114. Think About It Can every line be written in
slope-intercept form? Explain.
The Make a Decision exercise indicates a multipart exercise using large data sets. Go to this textbook’s
Companion Website to view these exercises.

1.2 Functions

Introduction to Functions What you should learn

● Decide whether a relation
Many everyday phenomena involve two quantities that are related to each other by
between two variables
some rule of correspondence. The mathematical term for such a rule of correspondence
represents a function.
is a relation. Here are two examples.
● Use function notation and
1. The simple interest I earned on an investment of $1000 for 1 year is related to the evaluate functions.
annual interest rate r by the formula I ⫽ 1000r. ● Find the domains of functions.
● Use functions to model and
2. The area A of a circle is related to its radius r by the formula A ⫽ ␲ r 2.
solve real-life problems.
Not all relations have simple mathematical formulas. For instance, people ● Evaluate difference quotients.
commonly match up NFL starting quarterbacks with touchdown passes, and time of day
with temperature. In each of these cases, there is some relation that matches each item
Why you should learn it
from one set with exactly one item from a different set. Such a relation is called a function. Many natural phenomena can be
modeled by functions, such as the
force of water against the face of a dam,
Definition of a Function explored in Exercise 78 on page 27.
A function f from a set A to a set B is a relation that assigns to each element x in
the set A exactly one element y in the set B. The set A is the domain (or set of
inputs) of the function f, and the set B contains the range (or set of outputs).

To help understand this definition, look at the function that relates the
time of day to the temperature in Figure 1.12.

Time of day (P.M.) Temperature (in degrees C)

1 9 2 1
2 133
4 4 5
5 15 6 7 8
6 3 12 14
10 16 11
Set A is the domain. Set B contains the range.
Inputs: 1, 2, 3, 4, 5, 6 Outputs: 9, 10, 12, 13, 15
Figure 1.12

This function can be represented by the ordered pairs

再共1, 9⬚兲, 共2, 13⬚兲, 共3, 15⬚兲, 共4, 15⬚兲, 共5, 12⬚兲, 共6, 10⬚兲冎.
In each ordered pair, the first coordinate (x-value) is the input and the second coordinate
( y-value) is the output.

Characteristics of a Function from Set A to Set B

1. Each element of A must be matched with an element of B.
2. Some elements of B may not be matched with any element of A.
Study Tip
3. Two or more elements of A may be matched with the same element of B. Be sure you see that
the range of a function
4. An element of A (the domain) cannot be matched with two different is not the same as the
elements of B. use of range relating to the
viewing window of a graphing
Andresr 2010/used under license from Shutterstock.com utility.

To determine whether or not a relation is a function, you must decide whether each
input value is matched with exactly one output value. When any input value is matched
with two or more output values, the relation is not a function.

Example 1 Testing for Functions

Decide whether the relation represents y as a function of x.
a. b. y
Input, x 2 2 3 4 5
3
Output, y 11 10 8 5 1 2
1
x
−3 −2 −1 1 2 3
−1
−2
−3

Figure 1.13

Solution
a. This table does not describe y as a function of x. The input value 2 is matched with
two different y-values.
b. The graph in Figure 1.13 does describe y as a function of x. Each input value is
matched with exactly one output value.
Now try Exercise 11.

In algebra, it is common to represent functions by equations or formulas involving

two variables. For instance, the equation y ⫽ x 2 represents the variable y as a function of
the variable x. In this equation, x is the independent variable and y is the dependent
variable. The domain of the function is the set of all values taken on by the independent
variable x, and the range of the function is the set of all values taken on by the dependent
variable y.

Example 2 Testing for Functions Represented Algebraically

Determine whether the equation represents y as a function of x. Explore the Concept
a. x2 ⫹y⫽1 b. ⫺x ⫹ y ⫽ 1 2
Use a graphing utility
to graph x 2 ⫹ y ⫽ 1.
Solution Then use the graph to
To determine whether y is a function of x, try to solve for y in terms of x. write a convincing argument
a. x 2 ⫹ y ⫽ 1 Write original equation. that each x-value corresponds
to at most one y-value.
y⫽1⫺ x2 Solve for y.
Use a graphing utility to graph
Each value of x corresponds to exactly one value of y. So, y is a function of x. ⫺x ⫹ y 2 ⫽ 1. (Hint: You will
b. ⫺x ⫹ y 2 ⫽ 1 Write original equation. need to use two equations.)
Does the graph represent y as
y2 ⫽ 1 ⫹ x Add x to each side. a function of x? Explain.
y ⫽ ±冪1 ⫹ x Solve for y.

The ± indicates that for a given value of x there correspond two values of y. For
instance, when x ⫽ 3, y ⫽ 2 or y ⫽ ⫺2. So, y is not a function of x.
Now try Exercise 23.

Function Notation
When an equation is used to represent a function, it is convenient to name the function
so that it can be referenced easily. For example, you know that the equation y ⫽ 1 ⫺ x 2
describes y as a function of x. Suppose you give this function the name “f.” Then you
can use the following function notation.
Input Output Equation
x f 共x兲 f 共x兲 ⫽ 1 ⫺ x 2
The symbol f 共x兲 is read as the value of f at x or simply f of x. The symbol f 共x兲
corresponds to the y-value for a given x. So, you can write y ⫽ f 共x兲. Keep in mind that
f is the name of the function, whereas f 共x兲 is the output value of the function at the input
value x. In function notation, the input is the independent variable and the output is the
dependent variable. For instance, the function f 共x兲 ⫽ 3 ⫺ 2x has function values
denoted by f 共⫺1兲, f 共0兲, and so on. To find these values, substitute the specified input
values into the given equation.
For x ⫽ ⫺1, f 共⫺1兲 ⫽ 3 ⫺ 2共⫺1兲 ⫽ 3 ⫹ 2 ⫽ 5.
For x ⫽ 0, f 共0兲 ⫽ 3 ⫺ 2共0兲 ⫽ 3 ⫺ 0 ⫽ 3.
Although f is often used as a convenient function name and x is often used as the
independent variable, you can use other letters. For instance,
f 共x兲 ⫽ x 2 ⫺ 4x ⫹ 7, f 共t兲 ⫽ t 2 ⫺ 4t ⫹ 7, and g共s兲 ⫽ s 2 ⫺ 4s ⫹ 7
all define the same function. In fact, the role of the independent variable is that of a
“placeholder.” Consequently, the function could be written as
f 共䊏兲 ⫽ 共䊏兲2 ⫺ 4共䊏兲 ⫹ 7.

Example 3 Evaluating a Function

Let g共x兲 ⫽ ⫺x 2 ⫹ 4x ⫹ 1. Find each value of the function.
a. g共2兲
b. g共t兲
c. g共x ⫹ 2兲

Solution
a. Replacing x with 2 in g共x兲 ⫽ ⫺x 2 ⫹ 4x ⫹ 1 yields the following.
g共2兲 ⫽ ⫺ 共2兲2 ⫹ 4共2兲 ⫹ 1 ⫽ ⫺4 ⫹ 8 ⫹ 1 ⫽ 5
b. Replacing x with t yields the following.
g共t兲 ⫽ ⫺ 共t兲2 ⫹ 4共t兲 ⫹ 1 ⫽ ⫺t 2 ⫹ 4t ⫹ 1
c. Replacing x with x ⫹ 2 yields the following.
g共x ⫹ 2兲 ⫽ ⫺ 共x ⫹ 2兲2 ⫹ 4共x ⫹ 2兲 ⫹ 1 Substitute x ⫹ 2 for x.

⫽ ⫺共 x2 ⫹ 4x ⫹ 4兲 ⫹ 4x ⫹ 8 ⫹ 1 Multiply.

⫽ ⫺x 2 ⫺ 4x ⫺ 4 ⫹ 4x ⫹ 8 ⫹ 1 Distributive Property

⫽ ⫺x 2 ⫹5 Simplify.

Now try Exercise 31.

In Example 3, note that g共x ⫹ 2兲 is not equal to g共x兲 ⫹ g共2兲. In general,

g共u ⫹ v兲 ⫽ g共u兲 ⫹ g共v兲.

Library of Parent Functions: Absolute Value Function

The parent absolute value function given by f 共x兲 ⫽ x can be written as a ⱍⱍ
piecewise-defined function. The basic characteristics of the parent absolute value
function are summarized below and on the inside cover of this text.

Graph of f 共x兲 ⫽ x ⫽ ⱍⱍ 冦x,⫺x, x ⱖ 0

x < 0
y

1
f(x) = ⏐x⏐
x
−2 −1 (0, 0) 2
−1

−2

Domain: 共⫺ ⬁, ⬁兲
Range: 关0, ⬁兲
Intercept: 共0, 0兲
Decreasing on 共⫺ ⬁, 0兲
Increasing on 共0, ⬁兲

A function defined by two or more equations over a specified domain is called a

piecewise-defined function.
Technology Tip
Example 4 A Piecewise–Defined Function Most graphing utilities
can graph piecewise-
Evaluate the function when x ⫽ ⫺1 and x ⫽ 0. defined functions.

冦xx ⫹⫺ 1,1,
2 x < 0 For instructions on how to enter
f 共x兲 ⫽ a piecewise-defined function
x ≥ 0
into your graphing utility,
Solution consult your user’s manual.
Because x ⫽ ⫺1 is less than 0, use f 共x兲 ⫽ x 2 ⫹ 1 to obtain You may find it helpful to set
your graphing utility to
f 共⫺1兲 ⫽ 共⫺1兲2 ⫹ 1 Substitute ⫺1 for x. dot mode before
⫽ 2. Simplify. graphing such
functions.
For x ⫽ 0, use f 共x兲 ⫽ x ⫺ 1 to obtain
f 共0兲 ⫽ 0 ⫺ 1 Substitute 0 for x.

⫽ ⫺1. Simplify.

The graph of f is shown in Figure 1.14.

4
x 2 + 1, x < 0
f(x) =
x − 1, x ≥ 0
−6 6

−4

Figure 1.14

Now try Exercise 39.

Stephane Bidouze 2010/used under license from Shutterstock.com

The Domain of a Function

The domain of a function can be described explicitly or it can be implied by Explore the Concept
the expression used to define the function. The implied domain is the set of all real
numbers for which the expression is defined. For instance, the function Use a graphing utility to
graph y ⫽ 冪4 ⫺ x2 .
1 Domain excludes x-values that What is the domain of
f 共x兲 ⫽
x2 ⫺ 4 result in division by zero. this function? Then graph
y ⫽ 冪x 2 ⫺ 4 . What is the
has an implied domain that consists of all real numbers x other than x ⫽ ± 2. These two
domain of this function? Do the
values are excluded from the domain because division by zero is undefined. Another
domains of these two functions
common type of implied domain is that used to avoid even roots of negative numbers.
overlap? If so, for what values?
For example, the function
Domain excludes x-values that result
f 共x兲 ⫽ 冪x in even roots of negative numbers.

is defined only for x ⱖ 0. So, its implied domain is the interval 关0, ⬁兲. In general, the
domain of a function excludes values that would cause division by zero or result in the
even root of a negative number.

Library of Parent Functions: Square Root Function

Radical functions arise from the use of rational exponents. The most common
radical function is the parent square root function given by f 共x兲 ⫽ 冪x. The basic
Study Tip
characteristics of the parent square root function are summarized below and on the Because the square root
inside cover of this text. function is not defined
Graph of f 共x兲 ⫽ 冪x y for x < 0, you must be
careful when analyzing the
Domain: 关0, ⬁兲 4
domains of complicated functions
Range: 关0, ⬁兲 3 f(x) = x involving the square root symbol.
Intercept: 共0, 0兲 2
Increasing on 共0, ⬁兲
1
x
−1 (0, 0) 2 3 4
−1

Example 5 Finding the Domain of a Function

Find the domain of each function.
a. f : 再共⫺3, 0兲, 共⫺1, 4兲, 共0, 2兲, 共2, 2兲, 共4, ⫺1兲冎
b. g共x兲 ⫽ ⫺3x2 ⫹ 4x ⫹ 5
1
c. h共x兲 ⫽
x⫹5

Solution
a. The domain of f consists of all first coordinates in the set of ordered pairs.
Domain ⫽ 再⫺3, ⫺1, 0, 2, 4冎
b. The domain of g is the set of all real numbers.
c. Excluding x-values that yield zero in the denominator, the domain of h is the set of
all real numbers x except x ⫽ ⫺5.

Now try Exercise 55.

Example 6 Finding the Domain of a Function

Find the domain of each function.
b. k共x兲 ⫽ 冪4 ⫺ 3x
4
a. Volume of a sphere: V ⫽ 3␲ r3

Solution
a. Because this function represents the volume of a sphere, the values of the radius r
must be positive (see Figure 1.15). So, the domain is the set of all real numbers r
such that r > 0.
r>0

Figure 1.15

b. This function is defined only for x-values for which 4 ⫺ 3x ⱖ 0. By solving this
inequality, you will find that the domain of k is all real numbers that are less than
4
or equal to 3.
Now try Exercise 61.

In Example 6(a), note that the domain of a function may be implied by the physical
4
context. For instance, from the equation V ⫽ 3␲ r 3 you would have no reason to restrict r
to positive values, but the physical context implies that a sphere cannot have a negative
or zero radius.
For some functions, it may be easier to find the domain and range of the function
by examining its graph.

Example 7 Finding the Domain and Range of a Function

Use a graphing utility to find the domain and range of the function f 共x兲 ⫽ 冪9 ⫺ x2.

Solution
Graph the function as y ⫽ 冪9 ⫺ x2, as shown in Figure 1.16. Using the trace feature
of a graphing utility, you can determine that the x-values extend from ⫺3 to 3 and
the y-values extend from 0 to 3. So, the domain of the function f is all real numbers
such that
⫺3 ⱕ x ⱕ 3 Domain of f

and the range of f is all real numbers such that

0 ⱕ y ⱕ 3. Range of f

f(x) = 9 − x2

−6 6

−2

Figure 1.16

Now try Exercise 65.

Applications

Example 8 Construction Employees

The number N (in millions) of employees in Construction Industry
the construction industry in the United States Employees
increased in a linear pattern from 2003 through N
2006 (see Figure 1.17). In 2007, the number

Number of employees (in millions)

8
dropped, then decreased through 2008 in a
7
different linear pattern. These two patterns
can be approximated by the function 6

冦 0.32t ⫹ 5.7, 3 ⱕ t ⱕ 6 5
N(t兲 ⫽
⫺0.42t ⫹ 10.5, 7 ⱕ t ⱕ 8 4

where t represents the year, with t ⫽ 3 3

corresponding to 2003. Use this function 2

to approximate the number of employees 1
for each year from 2003 to 2008.
t
(Source: U.S. Bureau of Labor Statistics) 3 4 5 6 7 8
Year (3 ↔ 2003)
Solution
Figure 1.17
From 2003 to 2006, use N共t兲 ⫽ 0.32t ⫹ 5.7.
6.66, 6.98, 7.3, 7.62
2003 2004 2005 2006

From 2007 to 2008, use N共t兲 ⫽ ⫺0.42t ⫹ 10.5.

7.56, 7.14
2007 2008

Now try Exercise 77.

Example 9 The Path of a Baseball

A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second
and an angle of 45⬚. The path of the baseball is given by the function
f 共x兲 ⫽ ⫺0.0032x 2 ⫹ x ⫹ 3
where f 共x兲 is the height of the baseball (in feet) and x is the horizontal distance from
home plate (in feet). Will the baseball clear a 10-foot fence located 300 feet from home
plate?

Algebraic Solution Graphical Solution

The height of the baseball is a function of the horizontal distance from 100

home plate. When x ⫽ 300, you can find the height of the baseball
as follows. When x = 300, y = 15.
So, the ball will clear
f 共x兲 ⫽ ⫺0.0032x2 ⫹ x ⫹ 3 Write original function. a 10-foot fence.
f 共300兲 ⫽ ⫺0.0032共300兲2 ⫹ 300 ⫹ 3 Substitute 300 for x. 0 400
0
⫽ 15 Simplify.

When x ⫽ 300, the height of the baseball is 15 feet. So, the baseball
will clear a 10-foot fence.
Now try Exercise 79.
DIGIcal/iStockphoto.com

Difference Quotients
One of the basic definitions in calculus employs the ratio
f 共x ⫹ h兲 ⫺ f 共x兲
, h ⫽ 0.
h
This ratio is called a difference quotient, as illustrated in Example 10.

Example 10 Evaluating a Difference Quotient

f 共x ⫹ h兲 ⫺ f 共x兲
For f 共x兲 ⫽ x 2 ⫺ 4x ⫹ 7, find .
h

Solution
f 共x ⫹ h兲 ⫺ f 共x兲关共x ⫹ h兲2 ⫺ 4共x ⫹ h兲 ⫹ 7兴 ⫺ 共x 2 ⫺ 4x ⫹ 7兲
⫽ Study Tip
h h
x 2 ⫹ 2xh ⫹ h 2 ⫺ 4x ⫺ 4h ⫹ 7 ⫺ x 2 ⫹ 4x ⫺ 7 Notice in Example 10
⫽ that h cannot be zero in
h
the original expression.
2xh ⫹ h 2 ⫺ 4h Therefore, you must restrict
⫽
h the domain of the simplified
h共2x ⫹ h ⫺ 4兲 expression by listing h ⫽ 0 so
⫽ that the simplified expression
h
is equivalent to the original
⫽ 2x ⫹ h ⫺ 4, h ⫽ 0 expression.

Now try Exercise 83.

Summary of Function Terminology

Function: A function is a relationship between two variables such that to each
value of the independent variable there corresponds exactly one value of the
dependent variable.

Function Notation: y ⫽ f 共x兲

f is the name of the function.
y is the dependent variable, or output value.
x is the independent variable, or input value.
f 共x兲 is the value of the function at x.

Domain: The domain of a function is the set of all values (inputs) of the
independent variable for which the function is defined. If x is in the domain
of f, then f is said to be defined at x. If x is not in the domain of f, then f is said
to be undefined at x.
Range: The range of a function is the set of all values (outputs) assumed by the
dependent variable (that is, the set of all function values).
Implied Domain: If f is defined by an algebraic expression and the domain is
not specified, then the implied domain consists of all real numbers for which the
expression is defined.

The symbol indicates an example or exercise that highlights algebraic techniques specifically
used in calculus.

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1.2 Exercises For instructions on how to use a graphing utility, see Appendix A.

Vocabulary and Concept Check

In Exercises 1 and 2, fill in the blanks.
1. A relation that assigns to each element x from a set of inputs, or _______ , exactly one
element y in a set of outputs, or _______ , is called a _______ .
2. For an equation that represents y as a function of x, the _______ variable is the set of all x
in the domain, and the _______ variable is the set of all y in the range.

3. Can the ordered pairs 共3, 0兲 and 共3, 5兲 represent a function?

4. To find g共x ⫹ 1兲, what do you substitute for x in the function g共x兲 ⫽ 3x ⫺ 2?
5. Does the domain of the function f 共x兲 ⫽ 冪1 ⫹ x include x ⫽ ⫺2?
6. Is the domain of a piecewise-defined function implied or explicitly described?

Procedures and Problem Solving

Testing for Functions In Exercises 7–10, does the Testing for Functions In Exercises 13 and 14, which sets
relation describe a function? Explain your reasoning. of ordered pairs represent functions from A to B? Explain.
7. Domain Range 8. Domain Range 13. A ⫽ 再0, 1, 2, 3冎 and B ⫽ 再⫺2, ⫺1, 0, 1, 2冎
−2 5 −2 3 (a) 再共0, 1兲, 共1, ⫺2兲, 共2, 0兲, 共3, 2兲冎
−1 6 −1 4 (b) 再共0, ⫺1兲, 共2, 2兲, 共1, ⫺2兲, 共3, 0兲, 共1, 1兲冎
0 7 0 5 (c) 再共0, 0兲, 共1, 0兲, 共2, 0兲, 共3, 0兲冎
1 8 1 14. A ⫽ 再a, b, c冎 and B ⫽ 再0, 1, 2, 3冎
2 2
(a) 再共a, 1兲, 共c, 2兲, 共c, 3兲, 共b, 3兲冎
9. Domain Range 10. Domain Range (b) 再共a, 1兲, 共b, 2兲, 共c, 3兲冎
Cubs (State) (Electoral votes (c) 再共1, a兲, 共0, a兲, 共2, c兲, 共3, b兲冎
National 2000–2010)
Pirates
League
Dodgers Alabama 3 Pharmacology In Exercises 15 and 16, use the graph,
Alaska 5 which shows the average prices of name brand and
Colorado 9 generic drug prescriptions in the United States.
American Orioles Delaware (Source: National Association of Chain Drug Stores)
League Yankees Nebraska y
Twins Vermont
120
Average price (in dollars)

Testing for Functions In Exercises 11 and 12, decide 100

whether the relation represents y as a function of x.
80
Explain your reasoning.
Name brand
60
11. Generic
Input, x ⫺3 ⫺1 0 1 3
40
Output, y ⫺9 ⫺1 0 1 9 20

x
12. 2000 2001 2002 2003 2004 2005 2006 2007
Input, x 0 1 2 1 0
Year
Output, y ⫺4 ⫺2 0 2 4
15. Is the average price of a name brand prescription a
function of the year? Is the average price of a generic
prescription a function of the year? Explain.
16. Let b共t兲 and g共t兲 represent the average prices of name
brand and generic prescriptions, respectively, in year t.
Find b共2007兲 and g共2000兲.

冦
Testing for Functions Represented Algebraically In x 2 ⫺ 4, x ⱕ 0
Exercises 17–28, determine whether the equation 42. f 共x兲 ⫽
1 ⫺ 2x 2, x > 0
represents y as a function of x.
(a) f 共⫺2兲 (b) f 共0兲 (c) f 共1兲
17. x2 ⫹ y2 ⫽ 4 18. x ⫽ y2 ⫹ 1

冦
x ⫹ 2, x < 0
19. y ⫽ 冪x2 ⫺ 1 20. y ⫽ 冪x ⫹ 5 43. f 共x兲 ⫽ 4, 0 ⱕ x < 2
21. 2x ⫹ 3y ⫽ 4 22. x ⫽ ⫺y ⫹ 5 x2 ⫹ 1, x ⱖ 2
23. y2 ⫽ x2 ⫺ 1 24. x ⫹ y2 ⫽ 3 (a) f 共⫺2兲 (b) f 共1兲 (c) f 共4兲
25. y⫽ 4⫺x ⱍ ⱍ 26. ⱍⱍ
y ⫽4⫺x

冦
5 ⫺ 2x, x < 0
27. x ⫽ ⫺7 28. y⫽8 44. f 共x兲 ⫽ 5, 0 ≤ x < 1
4x ⫹ 1, x ≥ 1
Evaluating a Function In Exercises 29– 44, evaluate the
(a) f 共⫺2兲 (b) f 共12兲 (c) f 共1兲
function at each specified value of the independent
variable and simplify.
Evaluating a Function In Exercises 45–48, assume that
29. f 共t兲 ⫽ 3t ⫹ 1 the domain of f is the set A ⴝ {ⴚ2, ⴚ1, 0, 1, 2}. Determine
(a) f 共2兲 (b) f 共⫺4兲 (c) f 共t ⫹ 2兲 the set of ordered pairs representing the function f.
30. g共 y兲 ⫽ 7 ⫺ 3y 45. f 共x兲 ⫽ x 2 46. f 共x兲 ⫽ x2 ⫺ 3
(a) g共0兲 (b) g共
7
3 兲 (c) g共s ⫹ 2兲 47. f 共x兲 ⫽ x ⫹ 2 ⱍⱍ 48. f 共x兲 ⫽ x ⫹ 1 ⱍ ⱍ
31. h共t兲 ⫽ t ⫺ 2t
2
Evaluating a Function In Exercises 49 and 50, complete
(a) h共2兲 (b) h共1.5兲 (c) h共x ⫹ 2兲
the table.
32. V共r兲 ⫽ 3␲ r
4 3

(a) V共3兲 (b) V 共 23 兲 (c) V 共2r兲

49. h共t兲 ⫽ 2 t ⫹ 3
1
ⱍ ⱍ
33. f 共 y兲 ⫽ 3 ⫺ 冪y t ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
(a) f 共4兲 (b) f 共0.25兲 (c) f 共4x 2兲
h共t兲
34. f 共x兲 ⫽ 冪x ⫹ 8 ⫹ 2
(a) f 共⫺4兲 f 共8兲 (c) f 共x ⫺ 8兲
1
(b)
50. f 共s兲 ⫽ ⱍs ⫺ 2ⱍ
35. q共x兲 ⫽ 2 s⫺2
x ⫺9
(a) q共⫺3兲 (b) q共2兲 (c) q共 y ⫹ 3兲 s 0 1 3
2
5
2 4
2t ⫹ 3
2
f 共s兲
36. q共t兲 ⫽
t2
(a) q共2兲 (b) q共0兲 (c) q共⫺x兲
Finding the Inputs That Have Outputs of Zero In
f 共x兲 ⫽ ⱍ ⱍ
x Exercises 51–54, find all values of x such that f 冇x冈 ⴝ 0.
37.
x
51. f 共x兲 ⫽ 15 ⫺ 3x 52. f 共x兲 ⫽ 5x ⫹ 1
(a) f 共9兲 (b) f 共⫺9兲 (c) f 共t兲
3x ⫺ 4 2x ⫺ 3
38. f 共x兲 ⫽ x ⫹ 4 ⱍⱍ 53. f 共x兲 ⫽
5
54. f 共x兲 ⫽
7
(a) f 共5兲 (b) f 共⫺5兲 (c) f 共t兲

冦
2x ⫹ 1, x < 0 Finding the Domain of a Function In Exercises 55–64,
39. f 共x兲 ⫽
2x ⫹ 2, x ⱖ 0 find the domain of the function.
(a) f 共⫺1兲 (b) f 共0兲 (c) f 共2兲 55. f 共x兲 ⫽ 5x 2 ⫹ 2x ⫺ 1 56. g共x兲 ⫽ 1 ⫺ 2x 2
xⱕ 0
冦
2x ⫹ 5, 4 3y
40. f 共x兲 ⫽ 57. h共t兲 ⫽ 58. s共 y兲 ⫽
2 ⫺ x2, x > 0 t y⫹5
(a) f 共⫺2兲 (b) f 共0兲 (c) f 共1兲 59. f 共x兲 ⫽ 冪3
x⫺4 60. f 共x兲 ⫽ 冪 x ⫹ 3x
4 2

冦
x ⫹ 2,
2 x ⱕ 1 1 3 10
41. f 共x兲 ⫽ 61. g共x兲 ⫽ ⫺ 62. h共x兲 ⫽ 2
2x 2 ⫹ 2, x > 1 x x⫹2 x ⫺ 2x
(a) f 共⫺2兲 (b) f 共1兲 (c) f 共2兲 y⫹2 冪x ⫹ 6
63. g共 y兲 ⫽ 64. f 共x兲 ⫽
冪y ⫺ 10 6⫹x

Finding the Domain and Range of a Function In 72. Geometry A right triangle is formed in the first
Exercises 65–68, use a graphing utility to graph the quadrant by the x- and y-axes and a line through the
function. Find the domain and range of the function. point 共2, 1兲共see figure兲. Write the area A of the triangle
as a function of x, and determine the domain of the
65. f 共x兲 ⫽ 冪4 ⫺ x2 66. f 共x兲 ⫽ 冪x2 ⫹ 1
function.
ⱍ
67. g共x兲 ⫽ 2x ⫹ 3 ⱍ 68. g共x兲 ⫽ x ⫺ 5 ⱍ ⱍ y
69. Geometry Write the area A of a circle as a function of
its circumference C. 4 (0, y)
70. Geometry Write the area A of an equilateral triangle 3
as a function of the length s of its sides.
2
71. Exploration An open box of maximum volume is to
be made from a square piece of material, 24 centimeters (2, 1)
1
on a side, by cutting equal squares from the corners and (x, 0)
turning up the sides (see figure). x
1 2 3 4
(a) The table shows the volume V (in cubic centimeters)
of the box for various heights x (in centimeters). 73. Geometry A rectangle is bounded by the x-axis and
Use the table to estimate the maximum volume. the semicircle y ⫽ 冪36 ⫺ x 2 (see figure). Write the
area A of the rectangle as a function of x, and determine
Height, x Volume, V the domain of the function.
y
1 484
2 800
3 972
8 y= 36 − x 2
4 1024
5 980 4 (x , y )
6 864
2
(b) Plot the points 共x, V兲 from the table in part (a). Does x
the relation defined by the ordered pairs represent V −6 −4 −2 2 4 6
as a function of x? −2
(c) If V is a function of x, write the function and determine
its domain. 74. Geometry A rectangular package to be sent by the
(d) Use a graphing utility to plot the points from the U.S. Postal Service can have a maximum combined
table in part (a) with the function from part (c). length and girth (perimeter of a cross section) of
How closely does the function represent the data? 108 inches (see figure).
Explain.
x

x 24 − 2x x
(a) Write the volume V of the package as a function of
x. What is the domain of the function?
x
(b) Use a graphing utility to graph the function. Be sure
to use an appropriate viewing window.
24 − 2x
(c) What dimensions will maximize the volume of the
package? Explain.

75. Business A company produces a handheld game 77. Civil Engineering The numbers n (in billions) of
console for which the variable cost is $68.20 per unit miles traveled by vans, pickup trucks, and sport utility
and the fixed costs are $248,000. The game console vehicles in the United States from 1990 through 2007
sells for $98.98. Let x be the number of units produced can be approximated by the model
and sold.
(a) The total cost for a business is the sum of the
variable cost and the fixed costs. Write the total cost
n共t兲 ⫽ 冦
⫺5.24t 2 ⫹ 69.5t ⫹ 581,
25.7t ⫹ 664,
0ⱕ tⱕ 6
6 < t ⱕ 17

C as a function of the number of units produced. where t represents the year, with t ⫽ 0 corresponding to
1990. The actual numbers are shown in the bar graph.
(b) Write the revenue R as a function of the number of
(Source: U.S. Federal Highway Administration)
units sold.
(c) Write the profit P as a function of the number of n
units sold. (Note: P ⫽ R ⫺ C.兲 1200

Miles traveled (in billions)

76. MODELING DATA 1000

The table shows the revenue y (in thousands 800

of dollars) of a landscaping business for 600
each month of 2010, with x ⫽ 1
representing January. 400

200
Month, x Revenue, y
t
1 5.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
2 5.6 Year (0 ↔ 1990)
3 6.6 (a) Identify the independent and dependent variables
4 8.3 and explain what they represent in the context of the
5 11.5 problem.
6 15.8 (b) Use the table feature of a graphing utility to
7 12.8 approximate the number of miles traveled by vans,
8 10.1 pickup trucks, and sport utility vehicles each year
9 8.6 from 1990 through 2007.
10 6.9 (c) Compare the values in part (b) with the actual
11 4.5 values shown in the bar graph. How well does the
12 2.7 model fit the data?
78. (p. 16) The force F (in tons)
The mathematical model below represents of water against the face of a dam is
the data. estimated by the function

f 共x兲 ⫽ 冦 ⫺1.97x ⫹ 26.3

0.505x2 ⫺ 1.47x ⫹ 6.3
F共 y兲 ⫽ 149.76冪10y 5兾2
where y is the depth of the water (in feet).
(a) Identify the independent and dependent variables (a) Complete the table. What can you
and explain what they represent in the context of conclude from it?
the problem.
(b) What is the domain of each part of the piecewise- y 5 10 20 30 40
defined function? Explain your reasoning.
F共 y兲
(c) Use the mathematical model to find f 共5兲. Interpret
your result in the context of the problem.
(b) Use a graphing utility to graph the function.
(d) Use the mathematical model to find f 共11兲. Describe your viewing window.
Interpret your result in the context of the problem.
(c) Use the table to approximate the depth at which the
(e) How do the values obtained from the models in parts force against the dam is 1,000,000 tons. Verify your
(c) and (d) compare with the actual data values? answer graphically. How could you find a better
estimate?
GLUE STOCK 2010/used under license from Shutterstock.com
Andresr 2010/used under license from Shutterstock.com

79. Projectile Motion A second baseman throws a baseball 1 f 共t兲 ⫺ f 共1兲

85. f 共t兲 ⫽ , , t⫽1
toward the first baseman 60 feet away. The path of the t t⫺1
ball is given by
4 f 共x兲 ⫺ f 共7兲
86. f 共x兲 ⫽ , , x⫽7
y ⫽ ⫺0.004x2 ⫹ 0.3x ⫹ 6 x⫹1 x⫺7
where y is the height (in feet) and x is the horizontal
distance (in feet) from the second baseman. The first
Conclusions
baseman can reach 8 feet high. Can the first baseman True or False? In Exercises 87 and 88, determine
catch the ball without jumping? Explain. whether the statement is true or false. Justify your answer.
80. Business The graph shows the sales (in millions of 87. The domain of the function f 共x兲 ⫽ x 4 ⫺ 1 is 共⫺ ⬁, ⬁兲,
dollars) of Peet’s Coffee & Tea from 2000 through and the range of f 共x兲 is 共0, ⬁兲.
2008. Let f 共x兲 represent the sales in year x. (Source:
88. The set of ordered pairs 再共⫺8, ⫺2兲, 共⫺6, 0兲, 共⫺4, 0兲,
Peet’s Coffee & Tea, Inc.)
共⫺2, 2兲, 共0, 4兲, 共2, ⫺2兲冎 represents a function.
f(x)
Think About It In Exercises 89 and 90, write a square
Sales (in millions of dollars)

300
root function for the graph shown. Then, identify the
250 domain and range of the function.
200 89. y 90. y

150 6
4
100 4
2
50 2
x
x x −4 −2 2
2000 2001 2002 2003 2004 2005 2006 2007 2008 −2 2 4 6 −2
Year −2

f 共2008兲 ⫺ f 共2000兲
(a) Find and interpret the result in 91. Think About It Given f 共x兲 ⫽ x2, is f the independent
2008 ⫺ 2000
the context of the problem. variable? Why or why not?
(b) An approximate model for the function is
92. C A P S T O N E
S共t兲 ⫽ 2.484t2 ⫹ 5.71t ⫹ 84.0, 0 ⱕ t ⱕ 8
(a) Describe any differences between a relation and a
where S is the sales (in millions of dollars) and function.
t ⫽ 0 represents 2000. Complete the table and (b) In your own words, explain the meanings of
compare the results with the data in the graph. domain and range.
t 0 1 2 3 4 5 6 7 8
S共t兲 Cumulative Mixed Review
Operations with Rational Expressions In Exercises
Evaluating a Difference Quotient In Exercises 81–86, 93–96, perform the operation and simplify.
find the difference quotient and simplify your answer. 4
93. 12 ⫺
f 共x ⫹ c兲 ⫺ f 共x兲 x⫹2
81. f 共x兲 ⫽ 2x, , c⫽0
c 3 x
94. ⫹ 2
g共x ⫹ h兲 ⫺ g 共x兲 x2 ⫹ x ⫺ 20 x ⫹ 4x ⫺ 5
82. g共x兲 ⫽ 3x ⫺ 1, , h⫽0
h 2x3 ⫹ 11x2 ⫺ 6x x ⫹ 10
95. ⭈ 2x2 ⫹ 5x ⫺ 3
f 共2 ⫹ h兲 ⫺ f 共2兲 5x
83. f 共x兲 ⫽ x2 ⫺ x ⫹ 1, , h⫽0
h x⫹7 x⫺7
96. ⫼
f 共x ⫹ h兲 ⫺ f 共x兲 2共x ⫺ 9兲 2共x ⫺ 9兲
84. f 共x兲 ⫽ x3 ⫹ x, , h⫽0
h

The symbol indicates an example or exercise that highlights algebraic techniques specifically
used in calculus.

1.3 Graphs of Functions

The Graph of a Function What you should learn

● Find the domains and ranges of
In Section 1.2, some functions were represented graphically by points on a graph in a
functions and use the Vertical
coordinate plane in which the input values are represented by the horizontal axis and
Line Test for functions.
the output values are represented by the vertical axis. The graph of a function f is the
● Determine intervals on which
collection of ordered pairs 共x, f 共x兲兲 such that x is in the domain of f. As you study this
functions are increasing,
section, remember the geometric interpretations of x and f 共x兲.
decreasing, or constant.
x ⫽ the directed distance from the y-axis ● Determine relative maximum
and relative minimum values of
f 共x兲 ⫽ the directed distance from the x-axis
functions.
Example 1 shows how to use the graph of a function to find the domain and range ● Identify and graph step functions
of the function. and other piecewise-defined
functions.
● Identify even and odd functions.
Example 1 Finding the Domain and Range of a Function
Use the graph of the function f shown in Figure 1.18 to find (a) the domain of f, (b) the
Why you should learn it
function values f 共⫺1兲 and f 共2兲, and (c) the range of f. Graphs of functions provide visual
relationships between two variables.
y For example, in Exercise 92 on page
(2, 4) 39, you will use the graph of a step
4
3
y = f(x) function to model the cost of sending
2
a package.
1
(4, 0)
x
1 2 3 4 5 6
Range

Domain
Figure 1.18

Solution
a. The closed dot at 共⫺1, ⫺5兲 indicates that x ⫽ ⫺1 is in the domain of f,
whereas the open dot at 共4, 0兲 indicates that x ⫽ 4 is not in the domain.
So, the domain of f is all x in the interval 关⫺1, 4兲.
b. Because 共⫺1, ⫺5兲 is a point on the graph of f, it follows that
f 共⫺1兲 ⫽ ⫺5.
Similarly, because 共2, 4兲 is a point on the graph of f, it follows that
f 共2兲 ⫽ 4.
c. Because the graph does not extend below f 共⫺1兲 ⫽ ⫺5 or above f 共2兲 ⫽ 4,
the range of f is the interval 关⫺5, 4兴.

Now try Exercise 9.

The use of dots (open or closed) at the extreme left and right points of a graph
indicates that the graph does not extend beyond these points. When no such dots are
shown, assume that the graph extends beyond these points.
Patrick Hermans 2010/used under license from Shutterstock.com

Example 2 Finding the Domain and Range of a Function

Find the domain and range of
f 共x兲 ⫽ 冪x ⫺ 4.

Algebraic Solution Graphical Solution

Because the expression under a radical cannot be negative, 5

the domain of f 共x兲 ⫽ 冪x ⫺ 4 is the set of all real numbers

such that y= x−4 The x-coordinates of points
on the graph extend from 4 to
x ⫺ 4 ⱖ 0. the right. So, the domain is
−1 8 the set of all real numbers
Solve this linear inequality for x as follows. (For help with greater than or equal to 4.
solving linear inequalities, see Appendix E at this textbook’s −1

Companion Website.) The y-coordinates of points

on the graph extend from 0
x⫺4 ⱖ 0 Write original inequality. upwards. So, the range is
the set of all nonnegative
x ⱖ 4 Add 4 to each side.
real numbers.
So, the domain is the set of all real numbers greater than or
equal to 4. Because the value of a radical expression is never
negative, the range of f 共x兲 ⫽ 冪x ⫺ 4 is the set of all
nonnegative real numbers.
Now try Exercise 13.

By the definition of a function, at most one y-value corresponds to a given

x-value. It follows, then, that a vertical line can intersect the graph of a function at most
once. This leads to the Vertical Line Test for functions.

Vertical Line Test for Functions

A set of points in a coordinate plane is the graph of y as a function of x if and
only if no vertical line intersects the graph at more than one point.

Example 3 Vertical Line Test for Functions

Use the Vertical Line Test to decide whether the graphs in Figure 1.19 represent y as a
function of x. Technology Tip
Most graphing utilities
4 4
are designed to graph
functions of x more
easily than other types of
−1 8 −2 7 equations. For instance, the
graph shown in Figure 1.19(a)
−2 −2
represents the equation
x ⫺ 共 y ⫺ 1兲2 ⫽ 0. To use a
(a) (b)
graphing utility to duplicate
Figure 1.19
this graph you must first solve
Solution the equation for y to obtain
y ⫽ 1 ± 冪x, and then graph
a. This is not a graph of y as a function of x because you can find a vertical line that the two equations y1 ⫽ 1 ⫹ 冪x
intersects the graph twice. and y2 ⫽ 1 ⫺ 冪x in the same
b. This is a graph of y as a function of x because every vertical line intersects the graph viewing window.
at most once.
Now try Exercise 21.

Increasing and Decreasing Functions

The more you know about the graph of a y

function, the more you know about the

4
function itself. Consider the graph shown

g
in Figure 1.20. Moving from left to right, 3

sin
cre as
this graph falls from x ⫽ ⫺2 to x ⫽ 0, is

rea
ing

Inc
constant from x ⫽ 0 to x ⫽ 2, and rises
Constant
from x ⫽ 2 to x ⫽ 4. 1

x
−2 −1 1 2 3 4
−1

Figure 1.20

Increasing, Decreasing, and Constant Functions

A function f is increasing on an interval when, for any x1 and x2 in the interval,
x1 < x2 implies f 共x1兲 < f 共x2兲.
A function f is decreasing on an interval when, for any x1 and x2 in the interval,
x1 < x2 implies f 共x1兲 > f 共x2兲.
A function f is constant on an interval when, for any x1 and x2 in the interval,
f 共x1兲 ⫽ f 共x2兲.

Example 4 Increasing and Decreasing Functions

In Figure 1.21, determine the open intervals on which each function is increasing,
decreasing, or constant.

x + 1, x < 0
f(x) = 1, 0≤x≤2
−x + 3 x > 2
2 f(x) = x 3 3 f(x) = x 3 − 3x 2

(−1, 2) (0, 1) (2, 1)

−3 3 −4 4 −2 4

(1, −2)
−2 −3 −2

(a) (b) (c)

Figure 1.21

Solution
a. Although it might appear that there is an interval in which this function is constant,
you can see that if x1 < x2, then 共x1兲3 < 共x2兲3, which implies that f 共x1兲 < f 共x2兲. So,
the function is increasing over the entire real line.
b. This function is increasing on the interval 共⫺ ⬁, ⫺1兲, decreasing on the interval
共⫺1, 1兲, and increasing on the interval 共1, ⬁兲.
c. This function is increasing on the interval 共⫺ ⬁, 0兲, constant on the interval 共0, 2兲,
and decreasing on the interval 共2, ⬁兲.
Now try Exercise 25.

Relative Minimum and Maximum Values

The points at which a function changes its increasing, decreasing, or constant behavior
are helpful in determining the relative maximum or relative minimum values of the
function.

y
Definition of Relative Minimum and Relative Maximum Relative
maxima
A function value f 共a兲 is called a relative minimum of f when there exists an
interval 共x1, x2兲 that contains a such that
x1 < x < x2 implies f 共a兲 ⱕ f 共x兲.
A function value f 共a兲 is called a relative maximum of f when there exists an
interval 共x1, x2兲 that contains a such that
Relative minima
x1 < x < x2 implies f 共a兲 ⱖ f 共x兲. x

Figure 1.22
Figure 1.22 shows several different examples of relative minima and relative
maxima. In Section 2.1, you will study a technique for finding the exact points at which
a second-degree polynomial function has a relative minimum or relative maximum. For
the time being, however, you can use a graphing utility to find reasonable approximations
of these points.

Example 5 Approximating a Relative Minimum

Use a graphing utility to approximate the relative minimum of the function given by
f 共x兲 ⫽ 3x2 ⫺ 4x ⫺ 2.

Solution
The graph of f is shown in Figure 1.23. By using the zoom and trace features of a
graphing utility, you can estimate that the function has a relative minimum at the point
Technology Tip
共0.67, ⫺3.33兲. See Figure 1.24.
When you use a graphing
Later, in Section 2.1, you will be able to determine that the exact point at which the utility to estimate the
relative minimum occurs is 共 3, ⫺ 3 兲.
2 10
x- and y-values of a
relative minimum or relative
2 f(x) = 3x 2 − 4x − 2 − 3.28
maximum, the zoom feature
will often produce graphs that
are nearly flat, as shown in
−4 5
Figure 1.24. To overcome this
problem, you can manually
change the vertical setting of
0.62 0.71
−4 − 3.39 the viewing window. The graph
will stretch vertically when the
Figure 1.23 Figure 1.24
values of Ymin and Ymax are
Now try Exercise 35. closer together.

Technology Tip
Some graphing utilities have built-in programs that will find minimum
or maximum values. These features are demonstrated in Example 6.

Example 6 Approximating Relative Minima and Maxima

Use a graphing utility to approximate the relative minimum and relative maximum of
the function given by f 共x兲 ⫽ ⫺x 3 ⫹ x.

Solution
By using the minimum and maximum features of the graphing utility, you can estimate
that the function has a relative minimum at the point
共⫺0.58, ⫺0.38兲 See Figure 1.25.

and a relative maximum at the point

共0.58, 0.38兲. See Figure 1.26.

If you take a course in calculus, you will learn a technique for finding the exact points
at which this function has a relative minimum and a relative maximum.

f(x) = −x 3 + x f(x) = −x 3 + x
2 2

−3 3 −3 3

−2 −2

Figure 1.25 Figure 1.26

Now try Exercise 37.

Example 7 Temperature
During a 24-hour period, the temperature y (in degrees Fahrenheit) of a
certain city can be approximated by the model
y ⫽ 0.026x3 ⫺ 1.03x2 ⫹ 10.2x ⫹ 34, 0 ⱕ x ⱕ 24
where x represents the time of day, with x ⫽ 0 corresponding to 6 A.M.
Approximate the maximum and minimum temperatures during this
24-hour period.

Solution
Using the maximum feature of a graphing utility, you can determine that the maximum
temperature during the 24-hour period was approximately 64⬚F. This temperature
occurred at about 12:36 P.M. 共x ⬇ 6.6兲, as shown in Figure 1.27. Using the minimum
feature, you can determine that the minimum temperature during the 24-hour period
was approximately 34⬚F, which occurred at about 1:48 A.M. 共x ⬇ 19.8兲, as shown in
Figure 1.28.

y = 0.026x 3 − 1.03x 2 + 10.2x + 34 y = 0.026x 3 − 1.03x 2 + 10.2x + 34

70 70

0 24 0 24
0 0
Figure 1.27 Figure 1.28
Now try Exercise 95.
ImageryMajestic 2010/used under license from Shutterstock.com

Step Functions and Piecewise-Defined Functions

Technology Tip
Library of Parent Functions: Greatest Integer Function Most graphing utilities
display graphs in
The greatest integer function, denoted by 冀x冁 and defined as the greatest
connected mode, which
integer less than or equal to x, has an infinite number of breaks or steps—
works well for graphs that do
one at each integer value in its domain. The basic characteristics of the greatest
not have breaks. For graphs
integer function are summarized below.
that do have breaks, such as the
Graph of f 共x兲 ⫽ 冀x冁 y f(x) = [[x]] greatest integer function, it is
Domain: 共⫺ ⬁, ⬁兲 3 better to use dot mode. Graph
Range: the set of integers 2 the greatest integer function
x-intercepts: in the interval 关0, 1兲 1 [often called Int 共x兲] in
y-intercept: 共0, 0兲 x connected and dot modes, and
−3 −2 1 2 3 compare the two results.
Constant between each pair of
consecutive integers
−3
Jumps vertically one unit at each
integer value

Because of the vertical jumps described above, the greatest integer function is an
example of a step function whose graph resembles a set of stairsteps. Some values of
the greatest integer function are as follows.
冀⫺1冁 ⫽ 共greatest integer ⱕ ⫺1兲 ⫽ ⫺1
冀⫺ 12冁 ⫽ 共greatest integer ⱕ ⫺ 12 兲 ⫽ ⫺1
冀101 冁 ⫽ 共greatest integer ⱕ 101 兲 ⫽ 0
What’s Wrong?
冀1.5冁 ⫽ 共greatest integer ⱕ 1.5兲 ⫽ 1
You use a graphing utility
In Section 1.2, you learned that a piecewise-defined function is a function that is
to graph
defined by two or more equations over a specified domain. To sketch the graph of a

冦x4 ⫹⫺ 1,x, x ⱕ 0
2
piecewise-defined function, you need to sketch the graph of each equation on the
f 共x兲 ⫽
appropriate portion of the domain. x > 0
by letting y1 ⫽ x2 ⫹ 1 and
Example 8 Sketching a Piecewise-Defined Function y2 ⫽ 4 ⫺ x, as shown in the
figure. You conclude that this is
Sketch the graph of
the graph of f. What’s wrong?
f 共x兲 ⫽ 冦⫺x2x ⫹⫹ 4,3, x ⱕ 1
x > 1
7

by hand.

Solution
−6 6
This piecewise-defined function is composed of
−1
two linear functions. At and to the left of x ⫽ 1,
the graph is the line given by
y ⫽ 2x ⫹ 3.
To the right of x ⫽ 1, the graph is the line given by
Figure 1.29
y ⫽ ⫺x ⫹ 4
as shown in Figure 1.29. Notice that the point 共1, 5兲 is a solid dot and the point 共1, 3兲
is an open dot. This is because f 共1兲 ⫽ 5.
Now try Exercise 55.

Even and Odd Functions

A graph has symmetry with respect to the y-axis if whenever 共x, y兲 is on the graph, then Explore the Concept
so is the point 共⫺x, y兲. A graph has symmetry with respect to the origin if whenever
共x, y兲 is on the graph, then so is the point 共⫺x, ⫺y兲. A graph has symmetry with respect Graph each function
to the x-axis if whenever 共x, y兲 is on the graph, then so is the point 共x, ⫺y兲. A function with a graphing utility.
whose graph is symmetric with respect to the y-axis is an even function. A function Determine whether the
whose graph is symmetric with respect to the origin is an odd function. A graph that is function is even, odd, or neither.
symmetric with respect to the x-axis is not the graph of a function 共except for the graph
f 共x兲 ⫽ x2 ⫺ x4
of y ⫽ 0兲. These three types of symmetry are illustrated in Figure 1.30.
g共x兲 ⫽ 2x3 ⫹ 1
y y y
h共x兲 ⫽ x5 ⫺ 2x3 ⫹ x
(x , y ) (x , y ) j共x兲 ⫽ 2 ⫺ x6 ⫺ x8
(−x, y) (x , y )
k共x兲 ⫽ x5 ⫺ 2x4 ⫹ x ⫺ 2

x x x p共x兲 ⫽ x9 ⫹ 3x5 ⫺ x3 ⫹ x
What do you notice about the
equations of functions that are
(−x, −y) (x, −y) (a) odd and (b) even? Describe
a way to identify a function as
(c) odd, (d) even, or (e) neither
Symmetric to y-axis Symmetric to origin Symmetric to x-axis
odd nor even by inspecting the
Even function Odd function Not a function
equation.
Figure 1.30

Example 9 Even and Odd Functions

Use the figure to determine whether the function is even, odd, or neither.

a. 4 b. 4

−6 6 −6 6

−4 −4

c. 4 d. 4

−6 6 −6 6

−4 −4

Solution
a. The graph is symmetric with respect to the y-axis. So, the function is even.
b. The graph is symmetric with respect to the origin. So, the function is odd.
c. The graph is neither symmetric with respect to the origin nor with respect to the
y-axis. So, the function is neither even nor odd.
d. The graph is symmetric with respect to the y-axis. So, the function is even.
Now try Exercise 67.
Andresr 2010/used under license from Shutterstock.com

Test for Even and Odd Functions

A function f is even when, for each x in the domain of f, f 共⫺x兲 ⫽ f 共x兲.
A function f is odd when, for each x in the domain of f, f 共⫺x兲 ⫽ ⫺f 共x兲.

Example 10 Even and Odd Functions

Determine whether each function is even, odd, or neither.
a. g共x兲 ⫽ x3 ⫺ x
b. h共x兲 ⫽ x2 ⫹ 1
c. f 共x兲 ⫽ x 3 ⫺ 1

Algebraic Solution Graphical Solution

a. This function is odd because a. In Figure 1.31, the graph is symmetric with respect to the
origin. So, this function is odd.
g共⫺x兲 ⫽ 共⫺x兲3 ⫺ 共⫺x兲
⫽ ⫺x3 ⫹ x 2

⫽ ⫺ 共x3 ⫺ x兲 (− x, − y) (x, y)
⫽ ⫺g共x兲. −3 3

b. This function is even because g(x) = x 3 − x

h共⫺x兲 ⫽ 共⫺x兲2 ⫹ 1 −2

Figure 1.31
⫽ x2 ⫹ 1
b. In Figure 1.32, the graph is symmetric with respect to the
⫽ h共x兲.
y-axis. So, this function is even.
c. Substituting ⫺x for x produces
3
f 共⫺x兲 ⫽ 共⫺x兲3 ⫺ 1
(− x, y) (x, y)
⫽ ⫺x3 ⫺ 1.
h(x) = x 2 + 1
Because −3 3
f 共x兲 ⫽ x ⫺ 1 3
−1
and
Figure 1.32
⫺f 共x兲 ⫽⫺x3 ⫹ 1
c. In Figure 1.33, the graph is neither symmetric with respect
you can conclude that to the origin nor with respect to the y-axis. So, this function
is neither even nor odd.
f 共⫺x兲 ⫽ f 共x兲
and 1

f 共⫺x兲 ⫽ ⫺f 共x兲. −3 3

So, the function is neither even nor odd.

f(x) = x 3 − 1

−3

Now try Exercise 81. Figure 1.33

To help visualize symmetry with respect to the origin, place a pin at the origin of
a graph and rotate the graph 180⬚. If the result after rotation coincides with the original
graph, then the graph is symmetric with respect to the origin.

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1.3 Exercises For instructions on how to use a graphing utility, see Appendix A.

Vocabulary and Concept Check

In Exercises 1 and 2, fill in the blank.
1. A function f is _______ on an interval when, for any x1 and x2 in the interval,
x1 < x2 implies f 共x1兲 > f 共x2兲.
2. A function f is _______ when, for each x in the domain of f, f 共⫺x兲 ⫽ f 共x兲.

3. The graph of a function f is the segment from 共1, 2兲 to 共4, 5兲, including the
endpoints. What is the domain of f ?
4. A vertical line intersects a graph twice. Does the graph represent a function?
5. Let f be a function such that f 共2兲 ⱖ f 共x兲 for all values of x in the interval 共0, 3兲.
Does f 共2兲 represent a relative minimum or a relative maximum?
6. Given f 共x兲 ⫽ 冀x冁, in what interval does f 共x兲 ⫽ 5?

Procedures and Problem Solving

Finding the Domain and Range of a Function In (d) What are the values of x from part (c) referred to
Exercises 7–10, use the graph of the function to find the graphically?
domain and range of f. Then find f 冇0冈. (e) Find f 冇0冈, if possible.
7. y 8. y (f) What is the value from part (e) referred to graphically?
3 y = f(x) 5 (g) What is the value of f at x ⴝ 1? What are the
2 y = f(x)
coordinates of the point?
x 2 (h) What is the value of f at x ⴝ ⴚ1? What are the
−2 −1 1 2 3 1 coordinates of the point?
−2 x
−3 −1 1 2 (i) The coordinates of the point on the graph of f at
−3
which x ⴝ ⴚ3 can be labeled 冇ⴚ3, f 冇ⴚ3冈冈, or
9. y 10. y 冇ⴚ3, 䊏冈.
6 y = f(x) 4 17. y 18. y

2 3 6
2 4
2 x
−2 2 4 1
x −2 x x
−2 2 4 y = f(x) −1 1 3 4 −4 −2 4 6
−2 −4 −2 −4
−3 −6
Finding the Domain and Range of a Function In f(x) = | x − 1 | − 2 x + 4, x ≤ 0
f(x) =
Exercises 11–16, use a graphing utility to graph the 4 − x 2, x > 0
function and estimate its domain and range. Then find
the domain and range algebraically.
Vertical Line Test for Functions In Exercises 19–22, use
11. f 共x兲 ⫽ 2x2 ⫹ 3 12. f 共x兲 ⫽ ⫺x2 ⫺ 1 the Vertical Line Test to determine whether y is a function
13. f 共x兲 ⫽ 冪x ⫺ 1 14. h共t兲 ⫽ 冪4 ⫺ t 2 of x. Describe how you can use a graphing utility to
15. f 共x兲 ⫽ x ⫹ 3 ⱍ ⱍ 16. f 共x兲 ⫽ ⫺ 4 x ⫺ 5
1
ⱍ ⱍ produce the given graph.
1
19. y ⫽ 2x 2 20. x ⫺ y 2 ⫽ 1
Analyzing a Graph In Exercises 17 and 18, use the 6 3
graph of the function to answer the questions.
(a) Determine the domain of the function. −1 8
(b) Determine the range of the function. −6 6

(c) Find the value(s) of x for which f 冇x冈 ⴝ 0. −2 −3

21. x 2 ⫹ y 2 ⫽ 25 22. x 2 ⫽ 2xy ⫺ 1 49. f 共x兲 ⫽ 冀x ⫺ 1冁 ⫹ 2 50. f 共x兲 ⫽ 冀x ⫺ 2冁 ⫹ 1

6 4 51. f 共x兲 ⫽ 冀2x冁 52. f 共x兲 ⫽ 冀4x冁

−9 9 −6 6 Describing a Step Function In Exercises 53 and 54, use

a graphing utility to graph the function. State the
domain and range of the function. Describe the pattern
−6 −4 of the graph.

Increasing and Decreasing Functions In Exercises 53. s共x兲 ⫽ 2共14x ⫺ 冀14x冁兲

54. g共x兲 ⫽ 2共14x ⫺ 冀14x冁兲
2
23–26, determine the open intervals on which the
function is increasing, decreasing, or constant.
Sketching a Piecewise-Defined Function In Exercises
23. f 共x兲 ⫽ 32x 24. f 共x兲 ⫽ x 2 ⫺ 4x
55–62, sketch the graph of the piecewise-defined function
4 3
by hand.

−6 6
−4 8
55. f 共x兲 ⫽ 冦2x3 ⫺⫹x,3, xx <ⱖ 00
56. f 共x兲 ⫽ 冦
x ⫹ 6, x ⱕ ⫺4
−4 −5
2x ⫺ 4, x > ⫺4
25. f 共x兲 ⫽ x3 ⫺ 3x 2 ⫹ 2 26. f 共x兲 ⫽ 冪x 2 ⫺ 1
57. f 共x兲 ⫽ 冦
4 ⫹ x, x < 0
冪
4 7 4 ⫺ x, x ⱖ 0
冪
1 ⫺ 共x ⫺ 1兲 , x ⱕ 2
58. f 共x兲 ⫽ 冦
2

−6 6 x ⫺ 2, 冪 x > 2

冦
−6 6 x ⫹ 3, x ⱕ 0
−4 −1 59. f 共x兲 ⫽ 3, 0 < x ⱕ 2
2x ⫺ 1, x > 2
Increasing and Decreasing Functions In Exercises

冦
x ⫹ 5, x ⱕ ⫺3
27–34, (a) use a graphing utility to graph the function
60. g共x兲 ⫽ ⫺2, ⫺3 < x < 1
and (b) determine the open intervals on which the
5x ⫺ 4, x ⱖ 1
function is increasing, decreasing, or constant.
27. f 共x兲 ⫽ 3 28. f 共x兲 ⫽ x 冦2x ⫹ 1,
61. f 共x兲 ⫽ 2
x ⫺ 2,
x ⱕ ⫺1
x > ⫺1
29. f 共x兲 ⫽ x 2兾3 30. f 共x兲 ⫽ ⫺x3兾4
62. h共x兲 ⫽ 冦
3 ⫹ x, x < 0
31. f 共x兲 ⫽ x冪x ⫹ 3 32. f 共x兲 ⫽ 冪1 ⫺ x x ⫹ 1, 2 x ⱖ 0
33. ⱍ
f 共x兲 ⫽ x ⫹ 1 ⫹ x ⫺ 1ⱍ ⱍ ⱍ
34. ⱍ ⱍ ⱍ ⱍ
f 共x兲 ⫽ ⫺ x ⫹ 4 ⫺ x ⫹ 1 Even and Odd Functions In Exercises 63–72, use a
graphing utility to graph the function and determine
Approximating Relative Minima and Maxima In whether it is even, odd, or neither.
Exercises 35–46, use a graphing utility to graph the 63. f 共x兲 ⫽ 5 64. f 共x兲 ⫽ ⫺9
function and to approximate any relative minimum or
65. f 共x兲 ⫽ 3x ⫺ 2 66. f 共x兲 ⫽ 5 ⫺ 3x
relative maximum values of the function.
67. h共x兲 ⫽ x2 ⫺ 4 68. f 共x兲 ⫽ ⫺x2 ⫺ 8
35. f 共x兲 ⫽ x 2 ⫺ 6x 36. f 共x兲 ⫽ 3x2 ⫺ 2x ⫺ 5
69. f 共x兲 ⫽ 冪1 ⫺ x 70. g共t兲 ⫽ 冪3 t ⫺ 1
37. y ⫽ 2x 3 ⫹ 3x 2 ⫺ 12x 38. y ⫽ x 3 ⫺ 6x 2 ⫹ 15
39. h共x兲 ⫽ 共x ⫺ 1兲冪x 40. g共x兲 ⫽ x冪4 ⫺ x
71. f 共x兲 ⫽ x ⫹ 2 ⱍ ⱍ 72. f 共x兲 ⫽ ⫺ x ⫺ 5 ⱍ ⱍ
41. f 共x兲 ⫽ x2 ⫺ 4x ⫺ 5 42. f 共x兲 ⫽ 3x2 ⫺ 12x Think About It In Exercises 73–78, find the coordinates
43. f 共x兲 ⫽ x ⫺ 3x 3 44. f 共x兲 ⫽ ⫺x ⫹ 3x 3 2 of a second point on the graph of a function f if the given
point is on the graph and the function is (a) even and
45. f 共x兲 ⫽ 3x2 ⫺ 6x ⫹ 1 46. f 共x兲 ⫽ 8x ⫺ 4x2
(b) odd.
Library of Parent Functions In Exercises 47–52, 73. 共⫺ 32, 4兲 74. 共⫺ 53, ⫺7兲
sketch the graph of the function by hand. Then use a 75. 共4, 9兲 76. 共5, ⫺1兲
graphing utility to verify the graph.
77. 共x, ⫺y兲 78. 共2a, 2c兲
47. f 共x兲 ⫽ 冀x冁 ⫹ 2 48. f 共x兲 ⫽ 冀x冁 ⫺ 3

Algebraic-Graphical-Numerical In Exercises 79–86, 95. MODELING DATA

determine whether the function is even, odd, or neither
The number N (in thousands) of existing condominiums
(a) algebraically, (b) graphically by using a graphing
and cooperative homes sold each year from 2000
utility to graph the function, and (c) numerically by
through 2008 in the United States is approximated by
using the table feature of the graphing utility to compare
the model
f 冇x冈 and f 冇ⴚx冈 for several values of x.
N ⫽ 0.4825t 4 ⫺ 11.293t 3 ⫹ 65.26t2 ⫺ 48.8t ⫹ 578,
79. f 共t兲 ⫽ t 2 ⫹ 2t ⫺ 3 80. f 共x兲 ⫽ x6 ⫺ 2x 2 ⫹ 3 0 ⱕ t ⱕ 8
81. g共x兲 ⫽ x 3 ⫺ 5x 82. h共x兲 ⫽ x 3 ⫺ 5
where t represents the year, with t ⫽ 0 corresponding
83. f 共x兲 ⫽ x冪1 ⫺ x 2 84. f 共x兲 ⫽ x冪x ⫹ 5 to 2000. (Source: National Association of Realtors)
85. g共s兲 ⫽ 4s 2兾3 86. f 共s兲 ⫽ 4s3兾2

Finding the Intervals Where a Function is Positive In

AVAVA 2010/used under license from Shutterstock.com

Exercises 87–90, graph the function and determine the

interval(s) (if any) on the real axis for which f 冇x冈 ⱖ 0.
Use a graphing utility to verify your results.
87. f 共x兲 ⫽ 4 ⫺ x 88. f 共x兲 ⫽ 4x ⫹ 2
89. f 共x兲 ⫽ x 2 ⫺ 9 90. f 共x兲 ⫽ x 2 ⫺ 4x

91. Business The cost of using a telephone calling card is

$1.05 for the first minute and $0.08 for each additional
minute or portion of a minute.
(a) A customer needs a model for the cost C of using
the calling card for a call lasting t minutes. Which (a) Use a graphing utility to graph the model over the
of the following is the appropriate model? appropriate domain.
(b) Use the graph from part (a) to determine during
C1共t兲 ⫽ 1.05 ⫹ 0.08冀t ⫺ 1冁 which years the number of cooperative homes and
C2共t兲 ⫽ 1.05 ⫺ 0.08冀⫺ 共t ⫺ 1兲冁 condos was increasing. During which years was
the number decreasing?
(b) Use a graphing utility to graph the appropriate (c) Approximate the maximum number of cooperative
Patrick Hermans 2010/used under license from Shutterstock.com

model. Estimate the cost of a call lasting 18 minutes homes and condos sold from 2000 through 2008.
and 45 seconds.
92. (p. 29) The cost of sending
an overnight package from New York to 96. Mechanical Engineering The intake pipe of a
Atlanta is $18.80 for a package weighing up 100-gallon tank has a flow rate of 10 gallons per minute,
to but not including 1 pound and $3.50 for and two drain pipes have a flow rate of 5 gallons per
each additional pound or portion of a minute each. The graph shows the volume V of fluid in
pound. Use the greatest integer function to the tank as a function of time t. Determine in which
create a model for the cost C of overnight pipes the fluid is flowing in specific subintervals of the
delivery of a package weighing x pounds, one-hour interval of time shown on the graph. (There
where x > 0. Sketch the graph of the function. are many correct answers.)
V
Using the Graph of a Function In Exercises 93 and 94,
(60, 100)
write the height h of the rectangle as a function of x. 100
Volume (in gallons)

93. y 94. y (10, 75) (20, 75)

75
4 y= − x2 + 4x − 1 4
(1, 3) (45, 50)
3 3 50
h h (5, 50) (50, 50)
2 2
(1, 2) (3, 2)
1 1 y = 4x − x 2 25
(30, 25) (40, 25)
x x (0, 0)
1 x 3 4 x1 2 3 4 t
10 20 30 40 50 60
Ti me (in minutes)

Conclusions 109. Let f be an even function. Determine whether g is

even, odd, or neither. Explain.
True or False? In Exercises 97 and 98, determine
(a) g共x兲 ⫽ ⫺f 共x兲 (b) g共x兲 ⫽ f 共⫺x兲
whether the statement is true or false. Justify your
answer. (c) g共x兲 ⫽ f 共x兲 ⫺ 2 (d) g共x兲 ⫽ ⫺f 共x ⫺ 2兲

97. A function with a square root cannot have a domain 110. C A P S T O N E Half of the graph of an odd function
that is the set of all real numbers. is shown.
98. It is possible for an odd function to have the interval (a) Sketch a complete graph y
关0, ⬁兲 as its domain. of the function.
2
Think About It In Exercises 99–104, match the graph (b) Find the domain and
1
of the function with the description that best fits the range of the function.
x
situation. (c) Identify the open intervals −2 −1 1 2
on which the function is −1
(a) The air temperature at a beach on a sunny day increasing, decreasing, or −2
(b) The height of a football kicked in a field goal attempt constant.
(c) The number of children in a family over time (d) Find any relative minimum
(d) The population of California as a function of time and relative maximum values
(e) The depth of the tide at a beach over a 24-hour period of the function.
(f) The number of cupcakes on a tray at a party
y y
111. Proof Prove that a function of the following form is
99. 100.
odd.
y ⫽ a2n⫹1x 2n⫹1 ⫹ a2n⫺1x 2n⫺1 ⫹ . . . ⫹ a3 x 3 ⫹ a1x
112. Proof Prove that a function of the following form is
x x even.
y ⫽ a2n x 2n ⫹ a 2n⫺2x 2n⫺2 ⫹ . . . ⫹ a2 x 2 ⫹ a 0

101. y 102. y Cumulative Mixed Review

Identifying Terms and Coefficients In Exercises
113 –116, identify the terms. Then identify the
coefficients of the variable terms of the expression.
x
113. ⫺2x2 ⫹ 8x 114. 10 ⫹ 3x
x
x
115. ⫺ 5x2 ⫹ x3 116. 7x 4 ⫹ 冪2x 2
3
103. y 104. y

Evaluating a Function In Exercises 117 and 118, evaluate

the function at each specified value of the independent
variable and simplify.
117. f 共x兲 ⫽ ⫺x2 ⫺ x ⫹ 3
x
x (a) f 共4兲 (b) f 共⫺2兲 (c) f 共x ⫺ 2兲
118. f 共x兲 ⫽ x冪x ⫺ 3
105. Think About It Does the graph in Exercise 20 (a) f 共3兲 (b) f 共12兲 (c) f 共6兲
represent x as a function of y? Explain.
106. Think About It Does the graph in Exercise 21 Evaluating a Difference Quotient In Exercises 119 and
represent x as a function of y? Explain. 120, find the difference quotient and simplify your
107. Think About It Can you represent the greatest answer.
integer function using a piecewise-defined function? f 共3 ⫹ h兲 ⫺ f 共3兲
119. f 共x兲 ⫽ x2 ⫺ 2x ⫹ 9, ,h⫽0
108. Think About It How does the graph of the greatest h
integer function differ from the graph of a line with a f 共6 ⫹ h兲 ⫺ f 共6兲
slope of zero? 120. f 共x兲 ⫽ 5 ⫹ 6x ⫺ x2, ,h⫽0
h

1.4 Shifting, Reflecting, and Stretching Graphs

Summary of Graphs of Parent Functions What you should learn

● Recognize graphs of parent
One of the goals of this text is to enable you to build your intuition for the basic shapes
functions.
of the graphs of different types of functions. For instance, from your study of lines
● Use vertical and horizontal shifts
in Section 1.1, you can determine the basic shape of the graph of the parent linear
and reflections to graph functions.
function
● Use nonrigid transformations to
f 共x兲 ⫽ x. graph functions.
Specifically, you know that the graph of this function is a line whose slope is 1 and Why you should learn it
whose y-intercept is 共0, 0兲. Recognizing the graphs of parent
The six graphs shown in Figure 1.34 represent the most commonly used types of functions and knowing how to
functions in algebra. Familiarity with the basic characteristics of these simple parent shift, reflect, and stretch graphs of
graphs will help you analyze the shapes of more complicated graphs. functions can help you sketch or
describe the graphs of a wide variety
Library of Parent Functions: Commonly Used Functions of simple functions. For example,
in Exercise 66 on page 49, you are
y y
asked to describe a transformation
2 2 that produces the graph of a model
for the sales of the WD-40 Company.
1 1
f(x) = x
x x
−2 −1 1 2 −2 −1 1 2
−1 −1 f(x) = x
−2 −2

(a) Linear Function (b) Absolute Value Function

y y

3 3
f(x) = x
2 2

1 1
f(x) = x 2
x x
−1 1 2 3 −2 −1 1 2
−1 −1

(c) Square Root Function (d) Quadratic Function

y y
1
2 2 f(x) =
x
1 1

x x
−2 −1 1 2 −1 1 2
−1 −1
f(x) = x 3
−2

(e) Cubic Function (f) Rational Function

Figure 1.34

Throughout this section, you will discover how many complicated graphs are
derived by shifting, stretching, shrinking, or reflecting the parent graphs shown above.
Shifts, stretches, shrinks, and reflections are called transformations. Many graphs of
functions can be created from combinations of these transformations.
Alessio Ponti 2010/used under license from Shutterstock.com Tyler Olson 2010/used under license from Shutterstock.com

Vertical and Horizontal Shifts

Many functions have graphs that are simple transformations of the graphs of parent
functions summarized in Figure 1.34. For example, you can obtain the graph of
Explore the Concept
h共x兲 ⫽ x 2 ⫹ 2
Use a graphing utility
by shifting the graph of f 共x兲 ⫽ x2 two units upward, as shown in Figure 1.35. In function
to display (in the same
notation, h and f are related as follows.
viewing window) the
h共x兲 ⫽ x2 ⫹ 2 graphs of y ⫽ x2 ⫹ c, where
c ⫽ ⫺2, 0, 2, and 4. Use the
⫽ f 共x兲 ⫹ 2 Upward shift of two units
results to describe the effect
Similarly, you can obtain the graph of that c has on the graph.
g共x兲 ⫽ 共x ⫺ 2兲2 Use a graphing utility to display
(in the same viewing window)
by shifting the graph of f 共x兲 ⫽ x2 two units to the right, as shown in Figure 1.36. In this the graphs of y ⫽ 共x ⫹ c兲2,
case, the functions g and f have the following relationship. where c ⫽ ⫺2, 0, 2, and 4. Use
g共x兲 ⫽ 共x ⫺ 2兲2 the results to describe the effect
that c has on the graph.
⫽ f 共x ⫺ 2兲 Right shift of two units

Vertical shift upward: two units Horizontal shift to the right: two units
Figure 1.35 Figure 1.36

The following list summarizes vertical and horizontal shifts.

Vertical and Horizontal Shifts

Let c be a positive real number. Vertical and horizontal shifts in the graph of
y ⫽ f 共x兲 are represented as follows.
1. Vertical shift c units upward: h共x兲 ⫽ f 共x兲 ⫹ c
2. Vertical shift c units downward: h共x兲 ⫽ f 共x兲 ⫺ c
3. Horizontal shift c units to the right: h共x兲 ⫽ f 共x ⫺ c兲
4. Horizontal shift c units to the left: h共x兲 ⫽ f 共x ⫹ c兲

In items 3 and 4, be sure you see that

h共x兲 ⫽ f 共x ⫺ c兲
corresponds to a right shift and
h共x兲 ⫽ f 共x ⫹ c兲
corresponds to a left shift for c > 0.

Example 1 Shifts in the Graph of a Function

Compare the graph of each function with the graph of f 共x兲 ⫽ x3.
a. g共x兲 ⫽ x3 ⫺ 1 b. h共x兲 ⫽ 共x ⫺ 1兲3 c. k共x兲 ⫽ 共x ⫹ 2兲3 ⫹ 1

Solution
a. You obtain the graph of g by shifting the graph of f one unit downward.
b. You obtain the graph of h by shifting the graph of f one unit to the right.
c. You obtain the graph of k by shifting the graph of f two units to the left and then one
unit upward.

(a) Vertical shift: one unit downward (b) Horizontal shift: one unit right (c) Two units left and one unit upward
Figure 1.37

Now try Exercise 23.

Example 2 Finding Equations from Graphs

The graph of f 共x兲 ⫽ x2 is shown in Figure 1.38. Each of the graphs in Figure 1.39 is a
transformation of the graph of f. Find an equation for each function.

(a) (b)
Figure 1.38 Figure 1.39

Solution
a. The graph of g is a vertical shift of four units upward of the graph of f 共x兲 ⫽ x2.
So, the equation for g is g共x兲 ⫽ x2 ⫹ 4.
b. The graph of h is a horizontal shift of two units to the left, and a vertical shift of one unit
downward, of the graph of f 共x兲 ⫽ x2. So, the equation for h is h共x兲 ⫽ 共x ⫹ 2兲2 ⫺ 1.
Now try Exercise 31.

Reflecting Graphs
Another common type of transformation is called a reflection. For instance, when you
consider the x-axis to be a mirror, the graph of h共x兲 ⫽ ⫺x2 is the mirror image (or
reflection) of the graph of f 共x兲 ⫽ x2 (see Figure 1.40). Explore the Concept
Compare the graph of
each function with the
graph of f 共x兲 ⫽ x2 by
using a graphing utility to graph
the function and f in the same
viewing window. Describe the
transformation.
a. g共x兲 ⫽ ⫺x2
b. h共x兲 ⫽ 共⫺x兲2

Figure 1.40

Reflections in the Coordinate Axes

Reflections in the coordinate axes of the graph of y ⫽ f 共x兲 are represented as follows.
1. Reflection in the x-axis: h共x兲 ⫽ ⫺f 共x兲
2. Reflection in the y-axis: h共x兲 ⫽ f 共⫺x兲

Example 3 Finding Equations from Graphs

The graph of f 共x兲 ⫽ x2 is shown in Figure 1.40. Each of the graphs in Figure 1.41 is a
transformation of the graph of f. Find an equation for each function.

(a) (b)
Figure 1.41

Solution
a. The graph of g is a reflection in the x-axis followed by an upward shift of two units
of the graph of f 共x兲 ⫽ x2. So, the equation for g is g共x兲 ⫽ ⫺x2 ⫹ 2.
b. The graph of h is a horizontal shift of three units to the right followed by a reflection
in the x-axis of the graph of f 共x兲 ⫽ x2. So, the equation for h is h共x兲 ⫽ ⫺ 共x ⫺ 3兲2.
Now try Exercise 33.
Edyta Pawlowska 2010/used under license from Shutterstock.com

Example 4 Reflections and Shifts

Compare the graph of each function with the graph of
f 共x兲 ⫽ 冪x.
a. g共x兲 ⫽ ⫺ 冪x
b. h共x兲 ⫽ 冪⫺x
c. k共x兲 ⫽ ⫺ 冪x ⫹ 2

Algebraic Solution Graphical Solution

a. Relative to the graph of f 共x兲 ⫽ 冪x, the a. From the graph in Figure 1.42,
graph of g is a reflection in the x-axis you can see that the graph of g
because is a reflection of the graph of f
in the x-axis. Note that the
g共x兲 ⫽ ⫺ 冪x
domain of g is x ⱖ 0.
⫽ ⫺f 共x兲.
b. The graph of h is a reflection of the graph
of f 共x兲 ⫽ 冪x in the y-axis because
h共x兲 ⫽ 冪⫺x
⫽ f 共⫺x兲.
c. From the equation
Figure 1.42
k共x兲 ⫽ ⫺ 冪x ⫹ 2
b. From the graph in Figure 1.43,
⫽ ⫺f 共x ⫹ 2兲 you can see that the graph of h
you can conclude that the graph of k is a is a reflection of the graph of f
left shift of two units, followed by a in the y-axis. Note that the
reflection in the x-axis, of the graph of domain of h is x ⱕ 0.
f 共x兲 ⫽ 冪x.

Figure 1.43

c. From the graph in Figure 1.44,

you can see that the graph of k
is a left shift of two units of the
graph of f, followed by a reflection
in the x-axis. Note that the domain
of k is x ⱖ ⫺2.

Now try Exercise 35. Figure 1.44

Nonrigid Transformations
Horizontal shifts, vertical shifts, and reflections are called rigid transformations
because the basic shape of the graph is unchanged. These transformations change only
the position of the graph in the coordinate plane. Nonrigid transformations are those
that cause a distortion—a change in the shape of the original graph. For instance, a
nonrigid transformation of the graph of y ⫽ f 共x兲 is represented by g共x兲 ⫽ cf 共x兲, where
the transformation is a vertical stretch when c > 1 and a vertical shrink when
0 < c < 1. Another nonrigid transformation of the graph of y ⫽ f 共x兲 is represented by
h共x兲 ⫽ f 共cx兲, where the transformation is a horizontal shrink when c > 1 and a
horizontal stretch when 0 < c < 1.

Example 5 Nonrigid Transformations

Compare the graph of each function with the graph of f 共x兲 ⫽ x . ⱍⱍ
a. h共x兲 ⫽ 3 x ⱍⱍ
b. g共x兲 ⫽ 3 x
1
ⱍⱍ
Solution
a. Relative to the graph of f 共x兲 ⫽ x , the graph of ⱍⱍ
ⱍⱍ
h共x兲 ⫽ 3 x ⫽ 3f 共x兲
is a vertical stretch (each y-value is multiplied by 3) of the graph of f. (See Figure 1.45.)
b. Similarly, the graph of

ⱍⱍ
g共x兲 ⫽ 13 x ⫽ 13 f 共x兲
is a vertical shrink 共each y-value is multiplied by 3 兲 of the graph of f. (See Figure 1.46.)
1

Figure 1.45 Figure 1.46

Now try Exercise 41.

Example 6 Nonrigid Transformations

Compare the graph of h共x兲 ⫽ f 共2 x兲 with the graph of f 共x兲 ⫽ 2 ⫺ x 3.
1

Solution
Relative to the graph of f 共x兲 ⫽ 2 ⫺ x3, the graph of
h共x兲 ⫽ f 共12 x兲 ⫽ 2 ⫺ 共12 x兲 ⫽ 2 ⫺ 18 x3
3

is a horizontal stretch (each x-value is multiplied by 2) of the graph of f. (See Figure 1.47.)
Now try Exercise 49. Figure 1.47

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1.4 Exercises For instructions on how to use a graphing utility, see Appendix A.

Vocabulary and Concept Check

1. Name three types of rigid transformations.
2. Match the rigid transformation of y ⫽ f 共x兲 with the correct representation, where c > 0.
(a) h共x兲 ⫽ f 共x兲 ⫹ c (i) horizontal shift c units to the left
(b) h共x兲 ⫽ f 共x兲 ⫺ c (ii) vertical shift c units upward
(c) h共x兲 ⫽ f 共x ⫺ c兲 (iii) horizontal shift c units to the right
(d) h共x兲 ⫽ f 共x ⫹ c兲 (iv) vertical shift c units downward

In Exercises 3 and 4, fill in the blanks.

3. A reflection in the x-axis of y ⫽ f 共x兲 is represented by h共x兲 ⫽ _______ ,
while a reflection in the y-axis of y ⫽ f 共x兲 is represented by h共x兲 ⫽ _______ .
4. A nonrigid transformation of y ⫽ f 共x兲 represented by cf 共x兲 is a vertical stretch
when _______ and a vertical shrink when _______ .

Procedures and Problem Solving

Sketching Transformations In Exercises 5–18, sketch Sketching Transformations In Exercises 19 and 20, use
the graphs of the three functions by hand on the same the graph of f to sketch each graph. To print an enlarged
rectangular coordinate system. Verify your results with a copy of the graph, go to the website www.mathgraphs.com.
graphing utility.
19. (a) y ⫽ f 共x兲 ⫹ 2 y

5. f 共x兲 ⫽ x 6. f 共x兲 ⫽ 2x
1
(b) y ⫽ ⫺f 共x兲 3
(4, 2)
g共x兲 ⫽ x ⫺ 4 g共x兲 ⫽ 12x ⫹ 2 (c) y ⫽ f 共x ⫺ 2兲
2
1 f (3, 1)
h共x兲 ⫽ 3x h共x兲 ⫽ 12共x ⫺ 2兲 (d) y ⫽ f 共x ⫹ 3兲 x
−2 −1
7. f 共x兲 ⫽ x 2 8. f 共x兲 ⫽ x 2 (e) y ⫽ 2f 共x兲
1 2 3 4
−2 (1, 0)
g共x兲 ⫽ x 2 ⫹ 2 g共x兲 ⫽ x 2 ⫺ 4 (f) y ⫽ f 共⫺x兲 (0, − 1)
−3
h共x兲 ⫽ 共x ⫺ 2兲2 h共x兲 ⫽ 共x ⫹ 2兲2 ⫹ 1 (g) y ⫽ f 共12 x兲
9. f 共x兲 ⫽ ⫺x 2 10. f 共x兲 ⫽ 共x ⫺ 2兲 2 20. (a) y ⫽ f 共x兲 ⫺ 1 y
(− 2, 4)
g共x兲 ⫽ ⫺x 2 ⫹ 1 g共x兲 ⫽ 共x ⫹ 2兲2 ⫹ 2 (b) y ⫽ f 共x ⫹ 1兲 4
h共x兲 ⫽ ⫺ 共x ⫺ 2兲2 h共x兲 ⫽ ⫺ 共x ⫺ 2兲 2 ⫺ 1 (c) y ⫽ f 共x ⫺ 1兲
f (0, 3)
2
11. f 共x兲 ⫽ x 2 12. f 共x兲 ⫽ x 2 (d) y ⫽ ⫺f 共x ⫺ 2兲 1
(1, 0)
g共x兲 ⫽ 12x2 g共x兲 ⫽ 14x2 ⫹ 2 (e) y ⫽ f 共⫺x兲 x
−3 −2 −1 1
h共x兲 ⫽ 共2x兲2 h共x兲 ⫽ ⫺ 14x2 (f) y ⫽ 12 f 共x兲 (3, −1)
−2
13. f 共x兲 ⫽ x ⱍⱍ 14. f 共x兲 ⫽ x ⱍⱍ (g) y ⫽ f 共2x兲
g共x兲 ⫽ x ⫺ 1ⱍⱍ g共x兲 ⫽ x ⫹ 3 ⱍ ⱍ
h共x兲 ⫽ ⱍx ⫺ 3ⱍ h共x兲 ⫽ ⫺2 x ⫹ 2 ⫺ 1 ⱍ ⱍ Error Analysis In Exercises 21 and 22, describe the
error in graphing the function.
15. f 共x兲 ⫽ 冪x 16. f 共x兲 ⫽ 冪x
g共x兲 ⫽ 冪x ⫹ 1 g共x兲 ⫽ 12冪x 21. f 共x兲 ⫽ 共x ⫹ 1兲2 22. f 共x兲 ⫽ 共x ⫺ 1兲2
y y
h共x兲 ⫽ 冪x ⫺ 2 ⫹ 1 h共x兲 ⫽ ⫺ 冪x ⫹ 4
6
1 1
17. f 共x兲 ⫽ 18. f 共x兲 ⫽ 4
x x
2
1 1 2
g共x兲 ⫽ ⫹ 2 g共x兲 ⫽ ⫺ 4
x x x x
−2 2 4 −2 2 4
1 1 −2 −2
h共x兲 ⫽ ⫹2 h共x兲 ⫽ ⫺1
x⫺1 x⫹3

Library of Parent Functions In Exercises 23–28, 49. f 共x兲 ⫽ x3 ⫺ 3x 2 50. f 共x兲 ⫽ x 3 ⫺ 3x 2 ⫹ 2

compare the graph of the function with the graph of its g共x兲 ⫽ ⫺ 13 f 共x兲 g共x兲 ⫽ ⫺f 共x兲
parent function. h共x兲 ⫽ f 共⫺x兲 h共x兲 ⫽ f 共2x兲
1
23. y ⫽ 冪x ⫹ 2 24. y ⫽ ⫺5 Describing Transformations In Exercises 51–64, g is
x
related to one of the six parent functions on page 41. (a)
25. y ⫽ 共x ⫺ 4兲3 26. y ⫽ x ⫹ 5 ⱍ ⱍ Identify the parent function f. (b) Describe the sequence
27. y ⫽ x2 ⫺ 2 28. y ⫽ 冪x ⫺ 2 of transformations from f to g. (c) Sketch the graph of g
by hand. (d) Use function notation to write g in terms of
Library of Parent Functions In Exercises 29–34, the parent function f.
identify the parent function and describe the transformation
51. g共x兲 ⫽ 2 ⫺ 共x ⫹ 5兲2 52. g共x兲 ⫽ ⫺ 共x ⫹ 10兲2 ⫹ 5
shown in the graph. Write an equation for the graphed
function. 53. g共x兲 ⫽ 3 ⫹ 2共x ⫺ 4兲2 54. g共x兲 ⫽ ⫺ 14共x ⫹ 2兲2 ⫺ 2
55. g共x兲 ⫽ 3共x ⫺ 2兲3 56. g共x兲 ⫽ ⫺ 12共x ⫹ 1兲3
29. 5 30. 4
57. g共x兲 ⫽ 共x ⫺ 1兲3 ⫹ 2
−8 4 58. g共x兲 ⫽ ⫺ 共x ⫹ 3兲3 ⫺ 10
−8 4
1 1
59. g共x兲 ⫽ ⫺9 60. g共x兲 ⫽ ⫹4
−3 −4
x⫹8 x⫺7

31. 2 32. 5 ⱍ ⱍ
61. g共x兲 ⫽ ⫺2 x ⫺ 1 ⫺ 4 62. g共x兲 ⫽ 12 x ⫺ 2 ⫺ 3 ⱍ ⱍ
63. g共x兲 ⫽ ⫺ 12冪x ⫹ 3 ⫺ 1 64. g共x兲 ⫽ ⫺ 冪x ⫹ 1 ⫺ 6
−3 3 65. MODELING DATA
−7 2 The amounts of fuel F (in billions of gallons) used by
−2 −1 motor vehicles from 1991 through 2007 are given by
the ordered pairs of the form 共t, F共t兲兲, where t ⫽ 1
33. 2 34. 3
represents 1991. A model for the data is
F共t兲 ⫽ ⫺0.099共t ⫺ 24.7兲2 ⫹ 183.4.
−1 5
−3 3 (Source: U.S. Federal Highway
−2 −1
Administration)
共1, 128.6兲
Rigid and Nonrigid Transformations In Exercises
共2, 132.9兲
35–46, compare the graph of the function with the graph
of its parent function. 共3, 137.3兲
共4, 140.8兲
35. y ⫽ ⫺ x ⱍⱍ 36. y ⫽ ⫺x ⱍ ⱍ 共5, 143.8兲
37. y ⫽ 共⫺x兲2 38. y ⫽ ⫺x3
共6, 147.4兲
1 1
39. y ⫽ 40. y ⫽ ⫺ 共7, 150.4兲
⫺x x
共8, 155.4兲
41. h共x兲 ⫽ 4 x ⱍⱍ 42. p共x兲 ⫽ 12x2
共9, 161.4兲
43. g共x兲 ⫽ 14x3 44. y ⫽ 2冪x
共10, 162.5兲共14, 173.5兲
45. f 共x兲 ⫽ 冪4x 46. y ⫽ 12 x ⱍ ⱍ 共11, 163.5兲共15, 174.8兲
Rigid and Nonrigid Transformations In Exercises 47–50, 共12, 168.7兲共16, 175.0兲
use a graphing utility to graph the three functions in the 共13, 170.0兲共17, 176.1兲
same viewing window. Describe the graphs of g and h
(a) Describe the transformation of the parent function
relative to the graph of f.
f 共t兲 ⫽ t2.
47. f 共x兲 ⫽ x3 ⫺ 3x 2 48. f 共x兲 ⫽ x 3 ⫺ 3x 2 ⫹ 2 (b) Use a graphing utility to graph the model and the
g共x兲 ⫽ f 共x ⫹ 2兲 g共x兲 ⫽ f 共x ⫺ 1兲 data in the same viewing window.
h共x兲 ⫽ 12 f 共x兲 h共x兲 ⫽ f 共3x兲 (c) Rewrite the function so that t ⫽ 0 represents 2000.
iofoto 2010/used under license from Shutterstock.com Explain how you got your answer.

75. y 76. y
66. (p. 41) The sales S (in
millions of dollars) of the WD-40 Company
from 2000 through 2008 can be approximated x
by the function
S共t兲 ⫽ 99冪t ⫹ 2.37 x

where t ⫽ 0 represents 2000. (Source:

WD-40 Company)
(a) Describe the transformation of the parent function (a) f 共x兲 ⫽ 共x ⫺ 2兲2 ⫺ 2 (a) f 共x兲 ⫽ ⫺ 共x ⫺ 4兲3 ⫹ 2
f 共t兲 ⫽ 冪t. (b) f 共x兲 ⫽ 共x ⫹ 4兲2 ⫺ 4 (b) f 共x兲 ⫽ ⫺ 共x ⫹ 4兲3 ⫹ 2
(b) Use a graphing utility to graph the model over the (c) f 共x兲 ⫽ 共x ⫺ 2兲2 ⫺ 4 (c) f 共x兲 ⫽ ⫺ 共x ⫺ 2兲3 ⫹ 4
interval 0 ⱕ t ⱕ 8. (d) f 共x兲 ⫽ 共x ⫹ 2兲2 ⫺ 4 (d) f 共x兲 ⫽ 共⫺x ⫺ 4兲3 ⫹ 2
(c) According to the model, in what year will the sales (e) f 共x兲 ⫽ 4 ⫺ 共x ⫺ 2兲2 (e) f 共x兲 ⫽ 共x ⫹ 4兲3 ⫹ 2
of WD-40 be approximately 400 million dollars?
(f) f 共x兲 ⫽ 4 ⫺ 共x ⫹ 2兲2 (f) f 共x兲 ⫽ 共⫺x ⫹ 4兲3 ⫹ 2
(d) Rewrite the function so that t ⫽ 0 represents 2005.
Explain how you got your answer. 77. Think About It You can use either of two methods to
graph a function: plotting points, or translating a parent
Conclusions function as shown in this section. Which method do you
prefer to use for each function? Explain.
True or False? In Exercises 67 and 68, determine (a) f 共x兲 ⫽ 3x2 ⫺ 4x ⫹ 1 (b) f 共x兲 ⫽ 2共x ⫺ 1兲2 ⫺ 6
whether the statement is true or false. Justify your
78. Think About It The graph of y ⫽ f 共x兲 passes through
answer.
the points 共0, 1兲, 共1, 2兲, and 共2, 3兲. Find the corresponding
67. The graph of y ⫽ f 共⫺x兲 is a reflection of the graph of points on the graph of y ⫽ f 共x ⫹ 2兲 ⫺ 1.
y ⫽ f 共x兲 in the x-axis. 79. Think About It Compare the graph of g共x兲 ⫽ ax2 with
68. The graphs of f 共x兲 ⫽ x ⫹ 6 and f 共x兲 ⫽ ⫺x ⫹ 6 are ⱍⱍ ⱍ ⱍ the graph of f 共x兲 ⫽ x2 when (a) 0 < a < 1 and (b)
identical. a > 1.

Exploration In Exercises 69–72, use the fact that the 80. C A P S T O N E Use the fact that the graph of y ⫽ f 共x兲
graph of y ⴝ f 冇x冈 has x-intercepts at x ⴝ 2 and x ⴝ ⴚ3 is increasing on the interval 共⫺ ⬁, 2兲 and decreasing
to find the x-intercepts of the given graph. If not possible, on the interval 共2, ⬁兲 to find the intervals on which
state the reason. the graph is increasing and decreasing. If not possible,
69. y ⫽ f 共⫺x兲 70. y ⫽ 2f 共x兲 state the reason.
71. y ⫽ f 共x兲 ⫹ 2 72. y ⫽ f 共x ⫺ 3兲 (a) y ⫽ f 共⫺x兲 (b) y ⫽ ⫺f 共x兲 (c) y ⫽ 2f 共x兲
(d) y ⫽ f 共x兲 ⫺ 3 (e) y ⫽ f 共x ⫹ 1兲
Library of Parent Functions In Exercises 73–76,
determine which equation(s) may be represented by the
graph shown. There may be more than one correct
Cumulative Mixed Review
answer.
73. y 74. y Parallel and Perpendicular Lines In Exercises 81 and 82,
determine whether the lines L1 and L2 passing through
x
the pairs of points are parallel, perpendicular, or neither.
81. L1: 共⫺2, ⫺2兲, 共2, 10兲
L2: 共⫺1, 3兲, 共3, 9兲
x
82. L1: 共⫺1, ⫺7兲, 共4, 3兲
L2: 共1, 5兲, 共⫺2, ⫺7兲
(a) ⱍ
f 共x兲 ⫽ x ⫹ 2 ⫹ 1 ⱍ (a) f 共x兲 ⫽ ⫺ 冪x ⫺ 4
(b) ⱍ
f 共x兲 ⫽ x ⫺ 1 ⫹ 2 ⱍ (b) f 共x兲 ⫽ ⫺4 ⫺ 冪x Finding the Domain of a Function In Exercises 83–86,
(c) ⱍ
f 共x兲 ⫽ x ⫺ 2 ⫹ 1 ⱍ (c) f 共x兲 ⫽ ⫺4 ⫺ 冪⫺x find the domain of the function.
(d) f 共x兲 ⫽ 2 ⫹ x ⫺ 2 ⱍ ⱍ (d) f 共x兲 ⫽ 冪⫺x ⫺ 4
83. f 共x兲 ⫽
4
84. f 共x兲 ⫽
冪x ⫺ 5
(e) ⱍ
f 共x兲 ⫽ 共x ⫺ 2兲 ⫹ 1 ⱍ (e) f 共x兲 ⫽ 冪⫺x ⫹ 4 9⫺x x⫺7
(f) f 共x兲 ⫽ 1 ⫺ x ⫺ 2 ⱍ ⱍ (f) f 共x兲 ⫽ 冪x ⫺ 4 85. f 共x兲 ⫽ 冪100 ⫺ x2 86. f 共x兲 ⫽ 冪
3
16 ⫺ x2
Alessio Ponti 2010/used under license from Shutterstock.com
Tyler Olson 2010/used under license from Shutterstock.com

1.5 Combinations of Functions

Arithmetic Combinations of Functions What you should learn

● Add, subtract, multiply,
Just as two real numbers can be combined by the operations of addition, subtraction,
and divide functions.
multiplication, and division to form other real numbers, two functions can be combined
● Find compositions of one
to create new functions. When
function with another function.
f 共x兲 ⫽ 2x ⫺ 3 and g共x兲 ⫽ x 2 ⫺ 1 ● Use combinations of functions
to model and solve real-life
you can form the sum, difference, product, and quotient of f and g as follows.
problems.
f 共x兲 ⫹ g共x兲 ⫽ 共2x ⫺ 3兲 ⫹ 共x 2 ⫺ 1兲
Why you should learn it
⫽ x 2 ⫹ 2x ⫺ 4 Sum You can model some situations by
f 共x兲 ⫺ g共x兲 ⫽ 共2x ⫺ 3兲 ⫺ 共x 2 ⫺ 1兲 combining functions. For instance,
in Exercise 79 on page 57, you will
⫽ ⫺x 2 ⫹ 2x ⫺ 2 Difference model the stopping distance of a car
f 共x兲 ⭈ g共x兲 ⫽ 共2x ⫺ 3兲共x ⫺ 1兲 2 by combining the driver’s reaction
time with the car’s braking distance.
⫽ 2x 3 ⫺ 3x 2 ⫺ 2x ⫹ 3 Product

f 共x兲 2x ⫺ 3
⫽ 2 , x ⫽ ±1 Quotient
g共x兲 x ⫺1
The domain of an arithmetic combination of functions f and g consists of all real
numbers that are common to the domains of f and g. In the case of the quotient
f 共x兲
g共x兲
there is the further restriction that g共x兲 ⫽ 0.

Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions with overlapping domains. Then, for all x
common to both domains, the sum, difference, product, and quotient of
f and g are defined as follows.
1. Sum: 共 f ⫹ g兲共x兲 ⫽ f 共x兲 ⫹ g共x兲
2. Difference: 共 f ⫺ g兲共x兲 ⫽ f 共x兲 ⫺ g共x兲
3. Product: 共 fg兲共x兲 ⫽ f 共x兲 ⭈ g共x兲
f 共x兲
冢g 冣共x兲 ⫽ g共x兲,
f
4. Quotient: g共x兲 ⫽ 0

Example 1 Finding the Sum of Two Functions

Given f 共x兲 ⫽ 2x ⫹ 1 and g共x兲 ⫽ x 2 ⫹ 2x ⫺ 1, find 共 f ⫹ g兲共x兲. Then evaluate the
sum when x ⫽ 2.

Solution
共 f ⫹ g兲共x兲 ⫽ f 共x兲 ⫹ g共x兲
⫽ 共2x ⫹ 1兲 ⫹ 共x 2 ⫹ 2x ⫺ 1兲
⫽ x2 ⫹ 4x
When x ⫽ 2, the value of this sum is 共 f ⫹ g兲共2兲 ⫽ 22 ⫹ 4共2兲 ⫽ 12.
Now try Exercise 13(a).
risteski goce 2010/used under license from Shutterstock.com
bignecker 2010/used under license from Shutterstock.com

Example 2 Finding the Difference of Two Functions

Given f 共x兲 ⫽ 2x ⫹ 1 and g共x兲 ⫽ x 2 ⫹ 2x ⫺ 1, find 共 f ⫺ g兲共x兲. Then evaluate the
difference when x ⫽ 2.

Algebraic Solution Graphical Solution

The difference of the functions f and g is Enter the functions in a graphing utility (see Figure 1.48). Then
graph the difference of the two functions, y3, as shown in
共 f ⫺ g兲共x兲 ⫽ f 共x兲 ⫺ g共x兲
Figure 1.49.
⫽ 共2x ⫹ 1兲 ⫺ 共x 2 ⫹ 2x ⫺ 1兲
⫽ ⫺x 2 ⫹ 2. 3
y3 = −x 2 + 2

When x ⫽ 2, the value of this difference is

共 f ⫺ g兲共2兲 ⫽ ⫺ 共2兲 2 ⫹ 2 −5 4

⫽ ⫺2.
−3
The value of
( f − g)(2) is − 2.
Now try Exercise 13(b). Figure 1.48 Figure 1.49

Example 3 Finding the Product of Two Functions

Given f 共x兲 ⫽ x2 and g共x兲 ⫽ x ⫺ 3, find 共 fg兲共x兲. Then evaluate the product when x ⫽ 4.

Solution
共 fg兲共x兲 ⫽ f 共x兲g 共x兲
⫽ 共x 2兲共x ⫺ 3兲
⫽ x3 ⫺ 3x 2
When x ⫽ 4, the value of this product is

共 fg兲共4兲 ⫽ 43 ⫺ 3共4兲2 ⫽ 16.

Now try Exercise 13(c).

In Examples 1–3, both f and g have domains that consist of all real
numbers. So, the domain of both 共 f ⫹ g兲 and 共 f ⫺ g兲 is also the set of all real numbers.
Remember that any restrictions on the domains of f or g must be considered when
forming the sum, difference, product, or quotient of f and g. For instance, the domain
of f 共x兲 ⫽ 1兾x is all x ⫽ 0, and the domain of g共x兲 ⫽ 冪x is 关0, ⬁兲. This implies that
the domain of 共 f ⫹ g兲 is 共0, ⬁兲.

Example 4 Finding the Quotient of Two Functions

Given f 共x兲 ⫽ 冪x and g共x兲 ⫽ 冪4 ⫺ x2, find 共 f兾g兲共x兲. Then find the domain of f兾g.

Solution

冢gf 冣共x兲 ⫽ gf 共共xx兲兲 ⫽ 冪4 ⫺x x

冪
2

The domain of f is 关0, ⬁兲 and the domain of g is 关⫺2, 2兴. The intersection of these
domains is 关0, 2兴. So, the domain of f兾g is 关0, 2兲.
Now try Exercise 13(d).

Compositions of Functions
Another way of combining two functions is to form the composition of one with the
other. For instance, when f 共x兲 ⫽ x 2 and g共x兲 ⫽ x ⫹ 1, the composition of f with g is
f 共g共x兲兲 ⫽ f 共x ⫹ 1兲
⫽ 共x ⫹ 1兲2.
This composition is denoted as f ⬚ g and is read as “f composed with g.”

Definition of Composition of Two Functions

The composition of the function f with the function g is
共 f ⬚ g兲共x兲 ⫽ f 共 g共x兲兲.
The domain of f ⬚ g is the set of all x in the domain of g such that g共x兲 is in the
domain of f. (See Figure 1.50.)

f °g

x g (x )
g f f (g(x))
Domain of g
Domain of f
Figure 1.50

Example 5 Forming the Composition of f with g

Find 共 f ⬚ g兲共x兲 for Explore the Concept
f 共x兲 ⫽ 冪x, x ⱖ 0, and g共x兲 ⫽ x ⫺ 1, x ⱖ 1. Let f 共x兲 ⫽ x ⫹ 2 and
If possible, find 共 f ⬚ g兲共2兲 and 共 f ⬚ g兲共0兲. g共x兲 ⫽ 4 ⫺ x 2. Are the
compositions f ⬚ g and
Solution g ⬚ f equal? You can use your
The composition of f with g is graphing utility to answer this
question by entering and graphing
共 f ⬚ g兲共x兲 ⫽ f 共 g共x兲兲 Definition of f ⬚ g the following functions.
⫽ f 共x ⫺ 1兲 Definition of g共x兲 y1 ⫽ 共4 ⫺ x 2兲 ⫹ 2
⫽ 冪x ⫺ 1, x ⱖ 1. Definition of f 共x兲 y2 ⫽ 4 ⫺ 共x ⫹ 2兲2
The domain of f ⬚ g is 关1, ⬁兲. (See Figure 1.51). So, What do you observe? Which
共 f ⬚ g兲共2兲 ⫽ 冪2 ⫺ 1 ⫽ 1 function represents f ⬚ g and
which represents g ⬚ f ?
is defined, but 共 f ⬚ g兲共0兲 is not defined because 0 is not in the domain of f ⬚ g.

3
( f ° g)(x) = x−1

−1 5

−1

Figure 1.51

Now try Exercise 41.

The composition of f with g is generally not the same as the composition of g

with f. This is illustrated in Example 6.

Example 6 Compositions of Functions

Given f 共x兲 ⫽ x ⫹ 2 and g共x兲 ⫽ 4 ⫺ x2, evaluate
(a) 共 f ⬚ g兲共x兲 and (b) 共g ⬚ f 兲共x兲
when x ⫽ 0 and 1.

Algebraic Solution Graphical Solution

a. and b. Enter y1 ⫽ f 共x), y2 ⫽ g共x兲, y3 ⫽ 共 f ⬚ g兲共x兲, and
a. 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲 Definition of f ⬚ g
y4 ⫽ 共g ⬚ f 兲共x兲, as shown in Figure 1.52. Then use
⫽ f 共4 ⫺ x 2兲 Definition of g共x兲 the table feature to find the desired function values
(see Figure 1.53).
⫽ 共4 ⫺ x 2兲 ⫹ 2 Definition of f 共x兲

⫽ ⫺x 2 ⫹6
共 f ⬚ g兲共0兲 ⫽ ⫺02 ⫹ 6 ⫽ 6
共 f ⬚ g兲共1兲 ⫽ ⫺12 ⫹ 6 ⫽ 5
b. 共g ⬚ f 兲共x兲 ⫽ g共 f (x)兲 Definition of g ⬚ f
Figure 1.52
⫽ g共x ⫹ 2兲 Definition of f 共x兲

⫽ 4 ⫺ 共x ⫹ 2兲2 Definition of g共x兲

⫽4⫺共 x2 ⫹ 4x ⫹ 4兲
⫽ ⫺x 2 ⫺ 4x
共g ⬚ f 兲共0兲 ⫽ ⫺02 ⫺ 4共0兲 ⫽ 0
共g ⬚ f 兲共1兲 ⫽ ⫺12 ⫺ 4共1兲 ⫽ ⫺5 Figure 1.53

Note that f ⬚ g ⫽ g ⬚ f. From the table you can see that f ⬚ g ⫽ g ⬚ f.

Now try Exercise 43.

Example 7 Finding the Domain of a Composite Function

Find the domain of f ⬚ g for the functions given by
f 共x兲 ⫽ x 2 ⫺ 9 and g共x兲 ⫽ 冪9 ⫺ x 2.

Algebraic Solution Graphical Solution

The composition of the functions is as follows. The x-coordinates of
points on the graph extend
共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲 from −3 to 3. So, the
⫽ f 共冪9 ⫺ x2 兲 domain of f ° g is [− 3, 3].

⫽ 共冪9 ⫺ x2 兲 ⫺ 9
2
2

−4 4
⫽ 9 ⫺ x2 ⫺ 9
⫽ ⫺x 2
From this, it might appear that the domain of the composition
is the set of all real numbers. This, however, is not true. −10
Because the domain of f is the set of all real numbers and the
domain of g is 关⫺3, 3兴, the domain of f ⬚ g is 关⫺3, 3兴.
Now try Exercise 45.

Example 8 A Case in Which f ⬚ g ⴝ g ⬚ f

Given
f 共x兲 ⫽ 2x ⫹ 3 and g共x兲 ⫽ 12共x ⫺ 3兲 Study Tip
find each composition. In Example 8, note
a. 共 f ⬚ g兲共x兲 that the two composite
functions f ⬚ g and
b. 共g ⬚ f 兲共x兲 g ⬚ f are equal, and both
Solution represent the identity function.
That is, 共 f ⬚ g兲共x兲 ⫽ x and
a. 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲共g ⬚ f 兲共x兲 ⫽ x. You will study

冢12 共x ⫺ 3兲冣
this special case in the next
⫽f section.

⫽2 冤 12 共x ⫺ 3兲冥 ⫹ 3
⫽x⫺3⫹3
⫽x
b. 共g ⬚ f 兲共x兲 ⫽ g共 f (x)兲
⫽ g共2x ⫹ 3兲
Explore the Concept
Write each function as
冤冥
1
⫽ 共2x ⫹ 3兲 ⫺ 3 a composition of two
2
functions.
1
⫽ 共2x兲
2 a. h共x兲 ⫽ x3 ⫺ 2 ⱍ ⱍ
⫽x
b. r共x兲 ⫽ x3 ⫺ 2 ⱍ ⱍ
What do you notice about the
Now try Exercise 57. inner and outer functions?
In Examples 5–8, you formed the composition of two given functions. In calculus,
it is also important to be able to identify two functions that make up a given composite
function. Basically, to “decompose” a composite function, look for an “inner” and an
“outer” function.

Example 9 Identifying a Composite Function

Write the function
1
h共x兲 ⫽
共x ⫺ 2兲 2
as a composition of two functions.

Solution
One way to write h as a composition of two functions is to take the inner function to be
g共x兲 ⫽ x ⫺ 2 and the outer function to be
1
f 共x兲 ⫽ ⫽ x⫺2.
x2
Then you can write
1
h共x兲 ⫽ ⫽ 共x ⫺ 2兲⫺2 ⫽ f 共x ⫺ 2兲 ⫽ f 共g共x兲兲.
共x ⫺ 2兲2
Now try Exercise 75.

Application

Example 10 Bacteria Count

The number N of bacteria in a refrigerated petri dish is given by
N共T 兲 ⫽ 20T 2 ⫺ 80T ⫹ 500, 2 ⱕ T ⱕ 14
where T is the temperature of the petri dish (in degrees Celsius). When the petri dish is
removed from refrigeration, the temperature of the petri dish is given by
T共t兲 ⫽ 4t ⫹ 2, 0 ⱕ t ⱕ 3
where t is the time (in hours).
a. Find the composition N共T共t兲兲 and interpret its meaning in context.
b. Find the number of bacteria in the petri dish when t ⫽ 2 hours.
c. Find the time when the bacteria count reaches 2000.

Solution
a. N共T共t兲兲 ⫽ 20共4t ⫹ 2兲2 ⫺ 80共4t ⫹ 2兲 ⫹ 500
⫽ 20共16t 2 ⫹ 16t ⫹ 4兲 ⫺ 320t ⫺ 160 ⫹ 500
⫽ 320t 2 ⫹ 320t ⫹ 80 ⫺ 320t ⫺ 160 ⫹ 500
⫽ 320t 2 ⫹ 420 Microbiologist
The composite function N共T共t兲兲 represents the number of bacteria as a function of
the amount of time the petri dish has been out of refrigeration.
b. When t ⫽ 2, the number of bacteria is
N ⫽ 320共2兲 2 ⫹ 420 ⫽ 1280 ⫹ 420 ⫽ 1700.
c. The bacteria count will reach N ⫽ 2000 when 320t 2 ⫹ 420 ⫽ 2000. You can solve
this equation for t algebraically as follows.
320t 2 ⫹ 420 ⫽ 2000
320t 2 ⫽ 1580
79
t2 ⫽
16
冪79
t⫽ t ⬇ 2.22 hours
4
So, the count will reach 2000 when t ⬇ 2.22 hours. Note that the negative value is rejected
because it is not in the domain of the composite function. To confirm your solution, graph
the equation N ⫽ 320t 2 ⫹ 420, as shown in Figure 1.54. Then use the zoom and trace
features to approximate N ⫽ 2000 when t ⬇ 2.22, as shown in Figure 1.55.

N = 320t 2 + 420, 2 ≤ t ≤ 3
3500 2500

2 3 2 3
1500 1500

Figure 1.54 Figure 1.55

Now try Exercise 85.

Leah-Anne Thompson 2010/used under license from Shutterstock.com

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1.5 Exercises For instructions on how to use a graphing utility, see Appendix A.

Vocabulary and Concept Check

In Exercises 1– 4, fill in the blank(s).
1. Two functions f and g can be combined by the arithmetic operations of _______ ,
_______ , _______ , and _______ to create new functions.
2. The _______ of the function f with the function g is 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲.
3. The domain of f ⬚ g is the set of all x in the domain of g such that _______ is in
the domain of f.
4. To decompose a composite function, look for an _______ and an _______ function.

5. Given f 共x兲 ⫽ x2 ⫹ 1 and 共 fg兲共x兲 ⫽ 2x共x2 ⫹ 1兲, what is g共x兲?

6. Given 共 f ⬚ g兲共x兲 ⫽ f 共x2 ⫹ 1兲, what is g共x兲?

Procedures and Problem Solving

Finding the Sum of Two Functions In Exercises 7–10, Evaluating an Arithmetic Combination of Functions In
use the graphs of f and g to graph h冇x冈 ⴝ 冇 f ⴙ g冈冇x冈. To Exercises 19–32, evaluate the indicated function for
print an enlarged copy of the graph, go to the website f 冇x冈 ⴝ x2 ⴚ 1 and g冇x冈 ⴝ x ⴚ 2 algebraically. If possible,
www.mathgraphs.com. use a graphing utility to verify your answer.
7. y 8. y 19. 共 f ⫹ g兲共3兲 20. 共 f ⫺ g兲共⫺2兲
3
f
3
g 21. 共 f ⫺ g兲共0兲 22. 共 f ⫹ g兲共1兲
2 2
1
23. 共 fg兲共6兲 24. 共 fg兲共⫺4兲
g f
x x 25. 共 f兾g 兲共⫺5兲 26. 共 f兾g 兲共0兲
− 2 −1 1 2 3 4 −3 −2 −1 2 3
27. 共 f ⫺ g兲共2t兲 28. 共 f ⫹ g兲共t ⫺ 4兲
−2 −2
−3 −3 29. 共 fg兲共⫺5t兲 30. 共 fg兲共3t2兲
y y
31. 共 f兾g 兲共⫺t兲 32. 共 f兾g兲共t ⫹ 2兲
9. 10.
5 3 Graphing an Arithmetic Combination of Functions In
4 g f
f Exercises 33–36, use a graphing utility to graph the
1
2 x
functions f, g, and h in the same viewing window.
g −3 −2 −1 1 3
f 共x兲 ⫽ 2 x, g共x兲 ⫽ x ⫺ 1, h共x兲 ⫽ f 共x兲 ⫹ g共x兲
1
33.
x −2
f 共x兲 ⫽ 3 x, g共x兲 ⫽ ⫺x ⫹ 4, h共x兲 ⫽ f 共x兲 ⫺ g共x兲
1
− 2 −1 1 2 3 4 −3
34.
35. f 共x兲 ⫽ x 2, g共x兲 ⫽ ⫺2x, h共x兲 ⫽ f 共x兲 ⭈ g共x兲
Finding Arithmetic Combinations of Functions In 36. f 共x兲 ⫽ 4 ⫺ x 2, g共x兲 ⫽ x, h共x兲 ⫽ f 共x兲兾g共x兲
Exercises 11–18, find (a) 冇 f ⴙ g冈冇x冈, (b) 冇 f ⴚ g冈冇x冈,
(c) 冇 fg冈冇x冈, and (d) 冇 f/g冈冇x冈. What is the domain of f/g? Graphing a Sum of Functions In Exercises 37– 40, use a
graphing utility to graph f, g, and f ⴙ g in the same
11. f 共x兲 ⫽
x ⫹ 3, g共x兲 ⫽ x ⫺ 3 viewing window. Which function contributes most to the
12. f 共x兲 ⫽
2x ⫺ 5, g共x兲 ⫽ 1 ⫺ x magnitude of the sum when 0 ⱕ x ⱕ 2? Which function
13. f 共x兲 ⫽
x 2, g共x兲 ⫽ 1 ⫺ x contributes most to the magnitude of the sum when
14. f 共x兲 ⫽
2x ⫺ 5, g共x兲 ⫽ 5 x > 6?
15. f 共x兲 ⫽
x 2 ⫹ 5, g共x兲 ⫽ 冪1 ⫺ x x3
37. f 共x兲 ⫽ 3x, g共x兲 ⫽ ⫺
x2 10
16. f 共x兲 ⫽ 冪x 2 ⫺ 4, g共x兲 ⫽ 2
x ⫹1 x
38. f 共x兲 ⫽ , g共x兲 ⫽ 冪x
1 1 2
17. f 共x兲 ⫽ , g共x兲 ⫽ 2
x x 39. f 共x兲 ⫽ 3x ⫹ 2, g共x兲 ⫽ ⫺ 冪x ⫹ 5
x 40. f 共x兲 ⫽ x2 ⫺ 12, g共x兲 ⫽ ⫺3x2 ⫺ 1
18. f 共x兲 ⫽ , g共x兲 ⫽ x 3
x⫹1

Compositions of Functions In Exercises 41– 44, find (a) 65. f 共x兲 ⫽ x , g共x兲 ⫽ 2x3ⱍⱍ
f ⬚ g, (b) g ⬚ f, and, if possible, (c) 冇 f ⬚ g冈冇0冈. 6
66. f 共x兲 ⫽ , g共x兲 ⫽ ⫺x
41. f 共x兲 ⫽ x2, g共x兲 ⫽ x ⫺ 1 3x ⫺ 5
42. f 共x兲 ⫽ 冪
3 x ⫺ 1, g共x兲 ⫽ x 3 ⫹ 1
Evaluating Combinations of Functions In Exercises
43. f 共x兲 ⫽ 3x ⫹ 5, g共x兲 ⫽ 5 ⫺ x
67–70, use the graphs of f and g to evaluate the functions.
1
44. f 共x兲 ⫽ x 3, g共x兲 ⫽ y
y = f(x)
y
x
4 4 y = g(x)
Finding the Domain of a Composite Function In 3 3
Exercises 45–54, determine the domains of (a) f, (b) g, 2 2
and (c) f ⬚ g. Use a graphing utility to verify your results.
1 1
45. f 共x兲 ⫽ 冪x ⫹ 4, g共x兲 ⫽ x2 x x
1 2 3 4 1 2 3 4
x
46. f 共x兲 ⫽ 冪x ⫹ 3, g(x) ⫽
2 67. (a) 共 f ⫹ g兲共3兲 (b) 共 f兾g兲共2兲
47. f 共x兲 ⫽ x2 ⫹ 1, g共x兲 ⫽ 冪x 68. (a) 共 f ⫺ g兲共1兲 (b) 共 fg兲共4兲
48. f 共x兲 ⫽ x1兾4 , g共x兲 ⫽ x4 69. (a) 共 f ⬚ g兲共2兲 (b) 共g ⬚ f 兲共2兲
49.
1
f 共x兲 ⫽ , g共x兲 ⫽ x ⫹ 3 70. (a) 共 f ⬚ g兲共1兲 (b) 共g ⬚ f 兲共3兲
x
1 1 Identifying a Composite Function In Exercises 71–78,
50. f 共x兲 ⫽ , g共x兲 ⫽ find two functions f and g such that 冇 f ⬚ g冈冇x冈 ⴝ h冇x冈.
x 2x
(There are many correct answers.)
51. f 共x兲 ⫽ x ⫺ 4 , ⱍ ⱍ g共x兲 ⫽ 3 ⫺ x
2 71. h共x兲 ⫽ 共2x ⫹ 1兲2 72. h共x兲 ⫽ 共1 ⫺ x兲3
52. f 共x兲 ⫽ , g共x兲 ⫽ x ⫺ 1 73. h共x兲 ⫽ 冪 74. h共x兲 ⫽ 冪9 ⫺ x
ⱍⱍ
3 x2 ⫺ 4
x
1 1
53. f 共x兲 ⫽ x ⫹ 2, g共x兲 ⫽ 75. h共x兲 ⫽
x2 ⫺4 x⫹2
3 4
54. f 共x兲 ⫽ , g共x兲 ⫽ x ⫹ 1 76. h共x兲 ⫽
x2 ⫺ 1 共5x ⫹ 2兲2
77. h共x兲 ⫽ 共x ⫹ 4兲 2 ⫹ 2共x ⫹ 4兲
Determining Whether f ⬚ g ⴝ g ⬚ f In Exercises 55–60, 78. h共x兲 ⫽ 共x ⫹ 3兲3兾2 ⫹ 4共x ⫹ 3兲1兾2
(a) find f ⬚ g, g ⬚ f, and the domain of f ⬚ g. (b) Use a
graphing utility to graph f ⬚ g and g ⬚ f. Determine 79. (p. 50) The research and
whether f ⬚ g ⴝ g ⬚ f. development department of an automobile
manufacturer has determined that when
55. f 共x兲 ⫽ 冪x ⫹ 4, g共x兲 ⫽ x 2
required to stop quickly to avoid an
56. f 共x兲 ⫽ 冪
3
x ⫹ 1, g共x兲 ⫽ x 3 ⫺ 1 accident, the distance (in feet) a car travels
f 共x兲 ⫽ 3 x ⫺ 3, g共x兲 ⫽ 3x ⫹ 9
1
57. during the driver’s reaction time is given by
58. f 共x兲 ⫽ 冪x, g共x兲 ⫽ 冪x R共x兲 ⫽ 34 x
59. f 共x兲 ⫽ x 2兾3, g共x兲 ⫽ x6
where x is the speed of the car in miles per hour. The
60. ⱍⱍ
f 共x兲 ⫽ x , g共x兲 ⫽ ⫺x2 ⫹ 1 distance (in feet) traveled while the driver is braking is
given by
Determining Whether f ⬚ g ⴝ g ⬚ f In Exercises 61–66,
B共x兲 ⫽ 15 x 2.
1
(a) find 冇 f ⬚ g冈冇x冈 and 冇 g ⬚ f 冈冇x冈, (b) determine
algebraically whether 冇 f ⬚ g冈冇x冈 ⴝ 冇 g ⬚ f 冈冇x冈, and (c) use a (a) Find the function that represents the total stopping
graphing utility to complete a table of values for the two distance T.
compositions to confirm your answer to part (b).
(b) Use a graphing utility to graph the functions R, B,
61. f 共x兲 ⫽ 5x ⫹ 4, g共x兲 ⫽ 4 ⫺ x and T in the same viewing window for 0 ⱕ x ⱕ 60.
f 共x兲 ⫽ 4共x ⫺ 1兲, g共x兲 ⫽ 4x ⫹ 1
1
62. (c) Which function contributes most to the magnitude
63. f 共x兲 ⫽ 冪x ⫹ 6, g共x兲 ⫽ x2 ⫺ 5 of the sum at higher speeds? Explain.
64. f 共x兲 ⫽ x3 ⫺ 4, g共x兲 ⫽ 冪3 x ⫹ 10

risteski goce 2010/used under license from Shutterstock.com

bignecker 2010/used under license from Shutterstock.com

80. MODELING DATA 82. Geometry A pebble is dropped into a calm pond,
causing ripples in the form of concentric circles. The
The table shows the total amounts (in billions of dollars)
radius (in feet) of the outermost ripple is given by
of private expenditures on health services and supplies
r 共t兲 ⫽ 0.6t, where t is the time (in seconds) after the
in the United States (including Puerto Rico) for the
pebble strikes the water. The area of the circle is given
years 1997 through 2007. The variables y1, y2, and y3
by A共r兲 ⫽ ␲ r 2. Find and interpret 共A ⬚ r兲共t兲.
represent out-of-pocket payments, insurance premiums,
and other types of payments, respectively. (Source: 83. Business A company owns two retail stores. The
U.S. Centers for Medicare and Medicaid Services) annual sales (in thousands of dollars) of the stores each
year from 2004 through 2010 can be approximated by
Year y1 y2 y3 the models

1997 162 359 52 S1 ⫽ 830 ⫹ 1.2t2 and S2 ⫽ 390 ⫹ 75.4t

1998 175 385 56 where t is the year, with t ⫽ 4 corresponding to 2004.
1999 184 417 59 (a) Write a function T that represents the total annual
2000 193 455 58 sales of the two stores.
2001 200 498 58 (b) Use a graphing utility to graph S1, S2, and T in the
2002 211 551 59 same viewing window.
2003 225 604 65 84. Business The annual cost C (in thousands of dollars)
2004 235 646 66 and revenue R (in thousands of dollars) for a company
2005 247 690 70 each year from 2004 through 2010 can be approximated
2006 255 731 75 by the models
2007 269 775 80 C ⫽ 260 ⫺ 8t ⫹ 1.6t2 and R ⫽ 320 ⫹ 2.8t
The data are approximated by where t is the year, with t ⫽ 4 corresponding to 2004.
the following models, where (a) Write a function P that represents the annual profits
t represents the year, with of the company.
t ⫽ 7 corresponding to 1997.
(b) Use a graphing utility to graph C, R, and P in the
y1 ⫽ 10.5t ⫹ 88 same viewing window.
y2 ⫽ 0.66t2 ⫹ 27.6t ⫹ 123 85. Biology The number of bacteria in a refrigerated food
product is given by
y3 ⫽ 0.23t2 ⫺ 3.0t ⫹ 64
N共T兲 ⫽ 10T 2 ⫺ 20T ⫹ 600, 1 ⱕ T ⱕ 20
(a) Use the models and the table feature of a graphing
utility to create a table showing the values of y1, y2, where T is the temperature of the food in degrees
and y3 for each year from 1997 through 2007. Celsius. When the food is removed from the refrigerator,
Compare these models with the original data. Are the temperature of the food is given by
the models a good fit? Explain. T共t兲 ⫽ 2t ⫹ 1
(b) Use the graphing utility to graph y1, y2, y3, and
where t is the time in hours.
yT ⫽ y1 ⫹ y2 ⫹ y3 in the same viewing window.
What does the function yT represent? (a) Find the composite function N共T共t兲兲 or 共N ⬚ T兲共t兲 and
interpret its meaning in the context of the situation.
(b) Find 共N ⬚ T兲共6兲 and interpret its meaning.
81. Geometry A square concrete foundation was
prepared as a base for a large cylindrical gasoline tank (c) Find the time when the bacteria count reaches 800.
Kurhan 2010/used under license from Shutterstock com.

(see figure). 86. Environmental Science The spread of a contaminant

is increasing in a circular pattern on the surface of a
(a) Write the radius r of the lake. The radius of the contaminant can be modeled by
tank as a function of the r共t兲 ⫽ 5.25冪t, where r is the radius in meters and t is
length x of the sides of time in hours since contamination.
the square. r
(a) Find a function that gives the area A of the circular
(b) Write the area A of the leak in terms of the time t since the spread began.
circular base of the tank
as a function of the (b) Find the size of the contaminated area after 36 hours.
radius r. (c) Find when the size of the contaminated area is
x
(c) Find and interpret 6250 square meters.
共A ⬚ r兲共x兲.

87. Air Traffic Control An air traffic controller spots two (b) The youngest sibling is two years old. Find the
planes flying at the same altitude. Their flight paths ages of the other two siblings.
form a right angle at point P. One plane is 150 miles
from point P and is moving at 450 miles per hour. The 93. Proof Prove that the product of two odd functions is
other plane is 200 miles from point P and is moving at an even function, and that the product of two even
450 miles per hour. Write the distance s between the functions is an even function.
planes as a function of time t. 94. Proof Use examples to hypothesize whether the
y product of an odd function and an even function is
even or odd. Then prove your hypothesis.
Distance (in miles)

200
95. Proof Given a function f, prove that g共x兲 is even and
h共x兲 is odd, where g共x兲 ⫽ 2 关 f 共x兲 ⫹ f 共⫺x兲兴 and
1

h共x兲 ⫽ 2 关 f 共x兲 ⫺ f 共⫺x兲兴.

1
s
100
96. (a) Use the result of Exercise 95 to prove that any
function can be written as a sum of even and odd
x functions. (Hint: Add the two equations in
P 100 200 Exercise 95.)
Distance (in miles)
(b) Use the result of part (a) to write each function as
88. Marketing The suggested retail price of a new car a sum of even and odd functions.
is p dollars. The dealership advertised a factory rebate 1
of $1200 and an 8% discount. f 共x兲 ⫽ x 2 ⫺ 2x ⫹ 1, g 共x兲 ⫽
x⫹1
(a) Write a function R in terms of p giving the cost of
the car after receiving the rebate from the factory. 97. Exploration The function in Example 9 can be
decomposed in other ways. For which of the following
(b) Write a function S in terms of p giving the cost of 1
pairs of functions is h共x兲 ⫽ equal to f 共g共x兲兲?
the car after receiving the dealership discount. 共x ⫺ 2兲2
(c) Form the composite functions 共R ⬚ S 兲共 p兲 and 1
共S ⬚ R兲共 p兲 and interpret each. (a) g共x兲 ⫽ and f 共x兲 ⫽ x2
x⫺2
(d) Find 共R ⬚ S兲共18,400兲 and 共S ⬚ R兲共18,400兲. Which 1
yields the lower cost for the car? Explain. (b) g共x兲 ⫽ x2 and f 共x兲 ⫽
x⫺2
Conclusions 1
(c) g共x兲 ⫽ 共x ⫺ 2兲2 and f 共x兲 ⫽
x
True or False? In Exercises 89 and 90, determine
whether the statement is true or false. Justify your answer. 98. C A P S T O N E Consider the functions f(x) ⫽ x2
89. A function that represents the graph of f 共x兲 ⫽ x2 and g共x兲 ⫽ 冪x. Describe the restrictions that need to
shifted three units to the right is f 共g共x兲兲, where be made on the domains of f and g so that
g共x兲 ⫽ x ⫹ 3. f 共g共x兲兲 ⫽ g共 f 共x兲兲.
90. Given two functions f 共x兲 and g共x兲, you can calculate
共 f ⬚ g兲共x兲 if and only if the range of g is a subset of the
domain of f. Cumulative Mixed Review
Evaluating an Equation In Exercises 99–102, find three
Exploration In Exercises 91 and 92, three siblings are points that lie on the graph of the equation. (There are
of three different ages. The oldest is twice the age of the many correct answers.)
middle sibling, and the middle sibling is six years older
than one-half the age of the youngest. 99. y ⫽ ⫺x2 ⫹ x ⫺ 5 100. y ⫽ 15 x3 ⫺ 4x2 ⫹ 1
x
101. x2 ⫹ y2 ⫽ 24 102. y ⫽ 2
91. (a) Write a composite function that gives the oldest x ⫺5
sibling’s age in terms of the youngest. Explain how Finding the Slope-Intercept Form In Exercises
you arrived at your answer. 103–106, find the slope-intercept form of the equation of
(b) The oldest sibling is 16 years old. Find the ages of the line that passes through the two points.
the other two siblings.
103. 共⫺4, ⫺2兲, 共⫺3, 8兲 104. 共1, 5兲, 共⫺8, 2兲
92. (a) Write a composite function that gives the youngest
sibling’s age in terms of the oldest. Explain how 105. 共32, ⫺1兲, 共⫺ 13, 4兲 106. 共0, 1.1兲, 共⫺4, 3.1兲
you arrived at your answer.

1.6 Inverse Functions

Inverse Functions What you should learn

● Find inverse functions informally
Recall from Section 1.2 that a function can be represented by a set of ordered pairs.
and verify that two functions are
For instance, the function f x ⫽ x ⫹ 4 from the set A ⫽ 1, 2, 3, 4 to the
inverse functions of each other.
set B ⫽ 5, 6, 7, 8 can be written as follows.
● Use graphs of functions to
f x ⫽ x ⫹ 4: 1, 5, 2, 6, 3, 7, 4, 8 decide whether functions have
inverse functions.
In this case, by interchanging the first and second coordinates of each of these ordered
● Determine whether functions
pairs, you can form the inverse function of f, which is denoted by f ⫺1. It is a function
are one-to-one.
from the set B to the set A, and can be written as follows.
● Find inverse functions
f ⫺1x ⫽ x ⫺ 4: 5, 1, 6, 2, 7, 3, 8, 4 algebraically.
Note that the domain of f is equal to the range of f ⫺1, and vice versa, as shown in Why you should learn it
Figure 1.56. Also note that the functions f and f ⫺1 have the effect of “undoing” each Inverse functions can be helpful in
other. In other words, when you form the composition of f with f ⫺1 or the composition further exploring how two variables
of f ⫺1 with f, you obtain the identity function. relate to each other. For example,
f f ⫺1x ⫽ f x ⫺ 4 ⫽ x ⫺ 4 ⫹ 4 ⫽ x in Exercise 115 on page 69, you
will use inverse functions to find the
f ⫺1 f x ⫽ f ⫺1x ⫹ 4 ⫽ x ⫹ 4 ⫺ 4 ⫽ x European shoe sizes from the
corresponding U.S. shoe sizes.
f (x) = x + 4
Domain of f Range of f

x f(x)

Range of f −1 Domain of f −1
−1
f (x) = x − 4

Figure 1.56

Example 1 Finding Inverse Functions Informally

Find the inverse function of
f(x) ⫽ 4x.
Then verify that both f f ⫺1x and f ⫺1 f x are equal to the identity function.

Solution
The function f multiplies each input by 4. To “undo” this function, you need to divide
each input by 4. So, the inverse function of f x ⫽ 4x is given by
x
f ⫺1x ⫽ .
4
You can verify that both f f ⫺1x and f ⫺1 f x are equal to the identity function as
follows.

4 ⫽ 4 4 ⫽ x
x x
f f ⫺1x ⫽ f

4x
f ⫺1 f x ⫽ f ⫺14x ⫽ ⫽x
4

Now try Exercise 7.

Andrey Armyagov 2010/used under license from Shutterstock.com

Don’t be confused by the use of the exponent ⫺1 to denote the inverse function
f ⫺1. In this text, whenever f ⫺1 is written, it always refers to the inverse function of the
function f and not to the reciprocal of f x, which is given by
1
.
f x

Example 2 Finding Inverse Functions Informally

Find the inverse function of f x ⫽ x ⫺ 6. Then verify that both f f ⫺1x and
f ⫺1 f x are equal to the identity function.

Solution
The function f subtracts 6 from each input. To “undo” this function, you need to add 6
to each input. So, the inverse function of f x ⫽ x ⫺ 6 is given by
f ⫺1x ⫽ x ⫹ 6.
You can verify that both f f ⫺1x and f ⫺1 f x are equal to the identity function as
follows.
f f ⫺1x ⫽ f x ⫹ 6 ⫽ x ⫹ 6 ⫺ 6 ⫽ x
f ⫺1 f x ⫽ f ⫺1x ⫺ 6 ⫽ x ⫺ 6 ⫹ 6 ⫽ x
Now try Exercise 9.

A table of values can help you understand inverse functions. For instance, the first
table below shows several values of the function in Example 2. Interchange the rows of
this table to obtain values of the inverse function.

x ⫺2 ⫺1 0 1 2 x ⫺8 ⫺7 ⫺6 ⫺5 ⫺4
f x ⫺8 ⫺7 ⫺6 ⫺5 ⫺4 f ⫺1 x ⫺2 ⫺1 0 1 2

In the table at the left, each output is 6 less than the input, and in the table at the right,
each output is 6 more than the input.
The formal definition of an inverse function is as follows.

Definition of Inverse Function

Let f and g be two functions such that
f gx ⫽ x for every x in the domain of g
and
g f x ⫽ x for every x in the domain of f.
Under these conditions, the function g is the inverse function of the function f.
The function g is denoted by f ⫺1 (read “f-inverse”). So,
f f ⫺1x ⫽ x and f ⫺1 f x ⫽ x.
The domain of f must be equal to the range of f ⫺1, and the range of f must be
equal to the domain of f ⫺1.

If the function g is the inverse function of the function f, then it must also be true
that the function f is the inverse function of the function g. For this reason, you can say
that the functions f and g are inverse functions of each other.

Example 3 Verifying Inverse Functions Algebraically

Show that the functions are inverse functions of each other. Technology Tip
f x ⫽ 2x3 ⫺ 1 and gx ⫽ x ⫹2 1
3 Most graphing utilities
can graph y ⫽ x1 3 in
two ways:
Solution
y1 ⫽ x 1 3 or
f gx ⫽ fx ⫹2 1 3
y1 ⫽
3 x.

⫽ 2
2
x⫹1 3
On some graphing utilities, you
⫺1
3
may not be able to obtain the
complete graph of y ⫽ x2 3 by
⫽2 x ⫹2 1 ⫺ 1 entering y1 ⫽ x 2 3. If not,
you should use
⫽x⫹1⫺1 y1 ⫽ x 1 3 2 or
⫽x y1 ⫽
3 x2 .

g f x ⫽ g2x3 ⫺ 1
y= 3

2x 3 ⫺ 1 ⫹ 1 x2 5
⫽ 3
2

⫽
3
2x
3
2 −6 6

⫽
3 x3
−3
⫽x
Now try Exercise 19.

Example 4 Verifying Inverse Functions Algebraically

5
Which of the functions is the inverse function of f x ⫽ ?
x⫺2
x⫺2 5
gx ⫽ or hx ⫽ ⫹2
5 x

Solution
By forming the composition of f with g, you have
x⫺2

5 25
f gx ⫽ f ⫽ ⫽ ⫽ x.
x⫺2 x ⫺ 12

5
⫺2
5
Because this composition is not equal to the identity function x, it follows that g is not
the inverse function of f. By forming the composition of f with h, you have

x ⫹ 2 ⫽
5 5 5
f hx ⫽ f ⫽ ⫽ x.

5 5
⫹2 ⫺2
x x
So, it appears that h is the inverse function of f. You can confirm this by showing
that the composition of h with f is also equal to the identity function.
Now try Exercise 23.

The Graph of an Inverse Function

The graphs of a function f and its inverse y
y=x Technology Tip
function f ⫺1 are related to each other in
the following way. If the point Many graphing utilities
y = f(x)
have a built-in feature
a, b for drawing an inverse
lies on the graph of f, then the point function. For instructions on
(a , b ) how to use the draw inverse
b, a y = f −1(x) feature, see Appendix A; for
must lie on the graph of f ⫺1, and vice versa. specific keystrokes,
(b , a ) go to this texbook’s
This means that the graph of f ⫺1 is a
reflection of the graph of f in the line Companion
y ⫽ x, as shown in Figure 1.57.
x Website.
Figure 1.57

Example 5 Verifying Inverse Functions Graphically

Verify that the functions f and g from Example 3 are inverse functions of each
other graphically.

Solution
From Figure 1.58, you can conclude x+1
that f and g are inverse functions of g(x) = 3
2 y =x
4
each other.
The graph of g is
a reflection of
the graph of f in −6 6
the line y = x.

−4
f(x) = 2x 3 − 1

Now try Exercise 33(b). Figure 1.58

Example 6 Verifying Inverse Functions Numerically

x⫺5
Verify that the functions f x ⫽ and gx ⫽ 2x ⫹ 5 are inverse functions of each
2
other numerically.

Solution
You can verify that f and g are inverse functions of each other numerically by using a
graphing utility. Enter y1 ⫽ f x, y2 ⫽ gx, y3 ⫽ f gx, and y4 ⫽ g f x, as shown in
Figure 1.59. Then use the table feature to create a table (see Figure 1.60).

Figure 1.59 Figure 1.60

Note that the entries for x, y3, and y4 are the same. So, f gx ⫽ x and g f x ⫽ x. You
can conclude that f and g are inverse functions of each other.
Now try Exercise 33(c).
Andresr 2010/used under license from Shutterstock.com

The Existence of an Inverse Function

To have an inverse function, a function must be one-to-one, which means that no two
elements in the domain of f correspond to the same element in the range of f.

Definition of a One-to-One Function

A function f is one-to-one when, for a and b in its domain, f a ⫽ f b implies
that a ⫽ b.

Existence of an Inverse Function

A function f has an inverse function f ⫺1 if and only if f is one-to-one.

From its graph, it is easy to tell whether a function of x is one-to-one. Simply check
to see that every horizontal line intersects the graph of the function at most once. This
is called the Horizontal Line Test. For instance, Figure 1.61 shows the graph of y ⫽ x2.
On the graph, you can find a horizontal line that intersects the graph twice.

y
y = x2
3

1
(−1, 1) (1, 1)
x
−2 −1 1 2

−1

Figure 1.61 f x ⴝ x2 is not one-to-one.

Two special types of functions that pass the Horizontal Line Test are those that are
increasing or decreasing on their entire domains.
1. If f is increasing on its entire domain, then f is one-to-one.
2. If f is decreasing on its entire domain, then f is one-to-one.

Example 7 Testing for One-to-One Functions

Is the function f x ⫽ x ⫹ 1 one-to-one?

Algebraic Solution Graphical Solution

Let a and b be nonnegative real numbers with f a ⫽ f b. y= x+1
A horizontal 5
a ⫹ 1 ⫽ b ⫹ 1 Set f a ⫽ f b. line will
intersect the
a ⫽ b The function
graph at most
is increasing.
a⫽b once.
−2 7
So, f a ⫽ f b implies that a ⫽ b. You can conclude that f is
−1
one-to-one and does have an inverse function.
Figure 1.62

From Figure 1.62, you can conclude that f is one-to-one and

Now try Exercise 67. does have an inverse function.

Finding Inverse Functions Algebraically

For simple functions, you can find inverse functions by inspection. For more complicated
functions, however, it is best to use the following guidelines.
What’s Wrong?
You use a graphing utility
Finding an Inverse Function
to graph y1 ⫽ x2 and then use
1. Use the Horizontal Line Test to decide whether f has an inverse function. the draw inverse feature to
conclude that f x ⫽ x2 has an
2. In the equation for f x, replace f x by y.
inverse function (see figure).
3. Interchange the roles of x and y, and solve for y. What’s wrong?
4. Replace y by f ⫺1x in the new equation.
5
5. Verify that f and f ⫺1 are inverse functions of each other by showing that
the domain of f is equal to the range of f ⫺1, the range of f is equal to
the domain of f ⫺1, and f f ⫺1x ⫽ x and f ⫺1 f x ⫽ x.
−6 6

−3
Example 8 Finding an Inverse Function Algebraically
Find the inverse function of
5 ⫺ 3x
f x ⫽ .
2

Solution
The graph of f in Figure 1.63 passes the Horizontal 3

Line Test. So, you know that f is one-to-one and has

an inverse function.
5 ⫺ 3x
f x ⫽ Write original function. −2 4
2
5 ⫺ 3x
−1 5 − 3x
f(x) =
y⫽ Replace f x by y. 2
2
Figure 1.63
5 ⫺ 3y
x⫽ Interchange x and y.
2
2x ⫽ 5 ⫺ 3y Multiply each side by 2.

3y ⫽ 5 ⫺ 2x Isolate the y-term.

5 ⫺ 2x
y⫽ Solve for y.
3
5 ⫺ 2x
f ⫺1x ⫽ Replace y by f ⫺1x.
3
The domains and ranges of f and f ⫺1 consist of all real numbers. Verify that
f f ⫺1x ⫽ x and f ⫺1 f x ⫽ x.

Now try Exercise 71.

A function f with an implied domain of all real numbers may not pass the
Horizontal Line Test. In this case, the domain of f may be restricted so that f does have
an inverse function. For instance, when the domain of f x ⫽ x2 is restricted to the
nonnegative real numbers, then f does have an inverse function.

Example 9 Finding an Inverse Function Algebraically

Find the inverse function of
f x ⫽ x3 ⫺ 4.

Solution
The graph of f in Figure 1.64 passes the Horizontal Line Test. So, you know that f is f(x) = x 3 − 4
one-to-one and has an inverse function. 4

f x ⫽ x3 ⫺ 4 Write original function.

−9 9
y⫽ x3 ⫺4 Replace f x by y.

x⫽ y3 ⫺4 Interchange x and y.
−8
y3 ⫽ x ⫹ 4 Isolate y.
Figure 1.64
y⫽
3 x ⫹ 4
Solve for y.

f ⫺1x ⫽
3 x ⫹ 4
Replace y by f ⫺1x.

The domains and ranges of f and f ⫺1 consist of all real numbers. You can verify that
f f ⫺1x ⫽ x and f ⫺1 f x ⫽ x as follows.
f f ⫺1x ⫽ f
3 x ⫹ 4
f ⫺1 f x ⫽ f ⫺1x3 ⫺ 4
⫽ ⫺4 3
3 x ⫹ 4
⫽
3
x3 ⫺ 4 ⫹ 4
⫽x⫹4⫺4 ⫽
3 3
x
⫽x ⫽x
Now try Exercise 73.

Example 10 Finding an Inverse Function Algebraically

Find the inverse function of
f x ⫽ 2x ⫺ 3.
5
f(x) = 2x − 3
Solution
The graph of f in Figure 1.65 passes the Horizontal Line Test. So, you know that f is
one-to-one and has an inverse function.
(32 , 0(
f x ⫽ 2x ⫺ 3 Write original function. −2 7

y ⫽ 2x ⫺ 3 Replace f x by y. −1

Figure 1.65
x ⫽ 2y ⫺ 3 Interchange x and y.

x 2 ⫽ 2y ⫺ 3 Square each side.

2y ⫽ x 2 ⫹ 3 Isolate y.

x2 ⫹3
y⫽ Solve for y.
2
x2 ⫹ 3
f ⫺1x ⫽ , x ⱖ 0 Replace y by f ⫺1x.
2
Note that the range of f is the interval 0, ⬁, which implies that the domain of f ⫺1 is
the interval 0, ⬁. Moreover, the domain of f is the interval 2, ⬁, which implies that
3

the range of f ⫺1 is the interval 2, ⬁. Verify that f f ⫺1x ⫽ x and f ⫺1 f x ⫽ x.
3

Now try Exercise 77.

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1.6 Exercises For instructions on how to use a graphing utility, see Appendix A.

Vocabulary and Concept Check

In Exercises 1–4, fill in the blank(s).
1. If f and g are functions such that f gx ⫽ x and g f x ⫽ x, then the function g is the
_______ function of f, and is denoted by _______ .
2. The domain of f is the _______ of f ⫺1, and the _______ of f ⫺1 is the range of f.
3. The graphs of f and f ⫺1 are reflections of each other in the line _______ .
4. To have an inverse function, a function f must be _______ ; that is, f a ⫽ f b implies a ⫽ b.

5. How many times can a horizontal line intersect the graph of a function that is one-to-one?
6. Can 1, 4 and 2, 4 be two ordered pairs of a one-to-one function?

Procedures and Problem Solving

Finding Inverse Functions Informally In Exercises 7–14, Verifying Inverse Functions Algebraically In Exercises
find the inverse function of f informally. Verify that 19–24, show that f and g are inverse functions
f f ⫺1 x ⴝ x and f ⫺1 f x ⴝ x. algebraically. Use a graphing utility to graph f and g in
the same viewing window. Describe the relationship
f x ⫽ 6x f x ⫽ 3 x
1
7. 8.
between the graphs.
9. f x ⫽ x ⫹ 7 10. f x ⫽ x ⫺ 3
1 1
11. f x ⫽ 2x ⫹ 1 12. f x ⫽ x ⫺ 1 4 19. f x ⫽ x 3, gx ⫽
20. f x ⫽ , gx ⫽
3 x
x x
13. f x ⫽
3 x 14. f x ⫽ x 5
21. f x ⫽ x ⫺ 4; gx ⫽ x 2 ⫹ 4, x ⱖ 0
Identifying Graphs of Inverse Functions In Exercises 22. f x ⫽ 9 ⫺ x 2, x ⱖ 0; gx ⫽ 9 ⫺ x
15–18, match the graph of the function with the graph of 23. f x ⫽ 1 ⫺ x 3, gx ⫽
3 1 ⫺ x

its inverse function. [The graphs of the inverse functions 1 1⫺x

are labeled (a), (b), (c), and (d).] 24. f x ⫽ , x ⱖ 0; gx ⫽ , 0 < x ⱕ 1
1⫹x x
(a) 7 (b) 7
Algebraic-Graphical-Numerical In Exercises 25–34,
show that f and g are inverse functions (a) algebraically,
(b) graphically, and (c) numerically.
−3 9 −3 9
−1 −1 7 2x ⫹ 6
25. f x ⫽ ⫺ x ⫺ 3, gx ⫽ ⫺
2 7
(c) 4 (d) 4
x⫺9
26. f x ⫽ , gx ⫽ 4x ⫹ 9
4
−6 6 −6 6
27. f x ⫽ x3 ⫹ 5, gx ⫽
3 x ⫺ 5

x3
−4 −4 28. f x ⫽ , gx ⫽
3
2x
2
15. 4 16. 7 29. f x ⫽ ⫺ x ⫺ 8, gx ⫽ 8 ⫹ x2, x ⱕ 0
x3 ⫹ 10
30. f x ⫽
3 3x ⫺ 10, gx ⫽
−6 6 3
−3 9 x
31. f x ⫽ 2x, gx ⫽
−4 −1 2
17. 7 18. 4 32. f x ⫽ x ⫺ 5, gx ⫽ x ⫹ 5
x⫺1 5x ⫹ 1
33. f x ⫽ , gx ⫽ ⫺
−6 6
x⫹5 x⫺1
x⫹3 2x ⫹ 3
−3 9 34. f x ⫽ , gx ⫽
−1 −4
x⫺2 x⫺1

Identifying Whether Functions Have Inverses In Exercises Analyzing a Piecewise-Defined Function In Exercises
35–38, does the function have an inverse? Explain. 57 and 58, sketch the graph of the piecewise-defined
function by hand and use the graph to determine
35. Domain Range 36. Domain Range
whether an inverse function exists.
1 can $1 1/2 hour $40
6 cans
12 cans
$5
$9
1 hour
2 hours
$70
$120
57. f x ⫽ x2, 0 ⱕ x ⱕ 1
x, x > 1

58. f x ⫽
24 cans $16 4 hours x ⫺ 2 , 3 x < 3
x ⫺ 4 , 2 x ⱖ 3
37. ⫺3, 6, ⫺1, 5, 0, 6
38. 2, 4, 3, 7, 7, 2 Testing for One-to-One Functions In Exercises 59–70,
determine algebraically whether the function is one-to-
Recognizing One-to-One Functions In Exercises 39–44, one. Verify your answer graphically. If the function is
determine whether the graph is that of a function. If so, one-to-one, find its inverse.
determine whether the function is one-to-one. 59. f x ⫽ x 4
39. y 40. y 60. gx ⫽ x 2 ⫺ x 4
3x ⫹ 4
61. f x ⫽
5
x 62. f x ⫽ 3x ⫹ 5
x
1
63. f x ⫽ 2
x
4
41. y 42. y 64. hx ⫽ 2
x
65. f x ⫽ x ⫹ 32, x ⱖ ⫺3
66. qx ⫽ x ⫺ 52, x ⱕ 5
67. f x ⫽ 2x ⫹ 3
x x 68. f x ⫽ x ⫺ 2

69. f x ⫽ x ⫺ 2 , x ⱕ 2
43. y 44. y x2
70. f x ⫽ 2
x ⫹1

Finding an Inverse Function Algebraically In Exercises

x 71–80, find the inverse function of f algebraically. Use a
x graphing utility to graph both f and f ⴚ1 in the same
viewing window. Describe the relationship between the
graphs.
Using the Horizontal Line Test In Exercises 45–56, use a 71. f x ⫽ 2x ⫺ 3
graphing utility to graph the function and use the 72. f x ⫽ 3x
Horizontal Line Test to determine whether the function 73. f x ⫽ x5
is one-to-one and thus has an inverse function.
74. f x ⫽ x3 ⫹ 1
45. f x ⫽ 3 ⫺ 12x 46. f x ⫽ 14x ⫹ 2 2 ⫺ 1 75. f x ⫽ x3 5
x2 4⫺x f x ⫽ x 2, x ⱖ 0
47. hx ⫽ 2 48. gx ⫽ 76.
x ⫹1 6x2
77. f x ⫽ 4 ⫺ x 2, 0 ⱕ x ⱕ 2
49. hx ⫽ 16 ⫺ x 2 50. f x ⫽ ⫺2x16 ⫺ x 2
78. f x ⫽ 16 ⫺ x2, ⫺4 ⱕ x ⱕ 0
51. f x ⫽ 10 52. f x ⫽ ⫺0.65
4
53. gx ⫽ x ⫹ 53 54. f x ⫽ x5 ⫺ 7 79. f x ⫽
x

55. hx ⫽ x ⫹ 4 ⫺ x ⫺ 4 80. f x ⫽
6
56. f x ⫽ ⫺
x⫺6 x
x⫹6

Think About It In Exercises 81–90, restrict the domain Using the Draw Inverse Feature In Exercises 101–104,
of the function f so that the function is one-to-one and (a) use a graphing utility to graph the function f, (b) use
has an inverse function. Then find the inverse function the draw inverse feature of the graphing utility to draw
f ⴚ1. State the domains and ranges of f and f ⴚ1. Explain the inverse relation of the function, and (c) determine
your results. (There are many correct answers.) whether the inverse relation is an inverse function.
Explain your reasoning.
81. f x ⫽ x ⫺ 2 2 82. f x ⫽ 1 ⫺ x 4
101. f x ⫽ x 3 ⫹ x ⫹ 1 102. f x ⫽ x4 ⫺ x 2
83. f x ⫽ x ⫹ 2 84. f x ⫽ x ⫺ 2
85. f x ⫽ x ⫹ 32 3x 2 4x
103. f x ⫽ 2 104. f x ⫽
86. f x ⫽ x ⫺ 42 x ⫹1 x 2 ⫹ 15

87. f x ⫽ ⫺2x2 ⫹ 5 Evaluating a Composition of Functions In Exercises

88. f x ⫽ 12x2 ⫺ 1 105–110, use the functions f x ⴝ 18 x ⴚ 3 and g x ⴝ x 3
89.
f x ⫽ x ⫺ 4 ⫹ 1 to find the indicated value or function.
90. f x ⫽ ⫺ x ⫺ 1 ⫺ 2 105. f ⫺1 ⬚ g⫺11 106. g⫺1 ⬚ f ⫺1⫺3
107. f ⫺1 ⬚ f ⫺16 108. g⫺1 ⬚ g⫺1⫺4
Using the Properties of Inverse Functions In Exercises
91 and 92, use the graph of the function f to complete the 109. f ⬚ g⫺1 110. g⫺1 ⬚ f ⫺1
table and sketch the graph of f ⫺1.
Finding a Composition of Functions In Exercises
y
91. x f ⫺1 x 111–114, use the functions f x ⴝ x ⴙ 4 and
4
g x ⴝ 2x ⴚ 5 to find the specified function.
⫺4
2 f 111. g⫺1 ⬚ f ⫺1 112. f ⫺1 ⬚ g⫺1
x ⫺2 113. f ⬚ g⫺1 114. g ⬚ f ⫺1
−4 −2 2 4
2
115. (p. 60) The table shows
3 men’s shoe sizes in the United States and
the corresponding European shoe sizes. Let
92. y y ⫽ f x represent the function that gives
x f ⫺1x
the men’s European shoe size in terms of x,
f
4 ⫺3 the men’s U.S. size.

⫺2
x
−4 −2 4 0 Men’s U.S. Men’s European
−2
shoe size shoe size
−4 6
8 41
9 42
Using Graphs to Evaluate a Function In Exercises 10 43
93–100, use the graphs of y ⴝ f x and y ⴝ g x to 11 45
evaluate the function. 12 46
y y 13 47
4 6
y = f(x) (a) Is f one-to-one? Explain.
y = g(x)
x
2 (b) Find f 11.
−4 −2 4
−6 −4
x
(c) Find f ⫺143, if possible.
−2 −2
2 4

−4
(d) Find f f ⫺141.
−4
(e) Find f ⫺1 f 13.
93. f ⫺10 94. g⫺10 116. Fashion Design Let y ⫽ gx represent the function
95. f ⬚ g2 96. g f ⫺4 that gives the women’s European shoe size in terms of
97. f ⫺1g0 98. g⫺1 ⬚ f 3 x, the women’s U.S. size. A women’s U.S. size 6 shoe
corresponds to a European size 38. Find g⫺1 g6.
99. g ⬚ f ⫺12 100. f ⫺1 ⬚ g⫺16
Andrey Armyagov 2010/used under license from Shutterstock.com

117. Military Science You can encode and decode 126. Think About It The domain of a one-to-one function
messages using functions and their inverses. To code a f is 0, 9 and the range is ⫺3, 3. Find the domain
message, first translate the letters to numbers using and range of f ⫺1.
1 for “A,” 2 for “B,” and so on. Use 0 for a space. So, 127. Think About It The function f x ⫽ 5x ⫹ 32 can be
9
“A ball” becomes used to convert a temperature of x degrees Celsius to
1 0 2 1 12 12. its corresponding temperature in degrees Fahrenheit.
(a) Using the expression for f, make a conceptual
Then, use a one-to-one function to convert to coded
argument to show that f has an inverse function.
numbers. Using f x ⫽ 2x ⫺ 1, “A ball” becomes
(b) What does f ⫺150 represent?
1 ⫺1 3 1 23 23. 128. Think About It A function f is increasing over its
(a) Encode “Call me later” using the function entire domain. Does f have an inverse function? Explain.
f x ⫽ 5x ⫹ 4. 129. Think About It Describe a type of function that is
(b) Find the inverse function of f x ⫽ 5x ⫹ 4 and not one-to-one on any interval of its domain.
use it to decode 119 44 9 104 4 104 49 69 29.
118. Production Management Your wage is $10.00 per 130. C A P S T O N E Decide whether the two functions
hour plus $0.75 for each unit produced per hour. So, shown in each graph appear to be inverse functions
your hourly wage y in terms of the number of units of each other. Explain your reasoning.
produced x is y ⫽ 10 ⫹ 0.75x. (a) y (b) y

(a) Find the inverse function. What does each variable 3 3

2 2
in the inverse function represent?
1
(b) Use a graphing utility to graph the function and its x x
inverse function. −3 −2 −1 2 3 −3 −2 2 3
−2
(c) Use the trace feature of the graphing utility to find the −3
hourly wage when 10 units are produced per hour.
(c) y (d) y
(d) Use the trace feature of the graphing utility to find
the number of units produced per hour when your 3
hourly wage is $21.25. 2 2
1
x x
Conclusions −1 2 3 −2 −1 1 2

True or False? In Exercises 119 and 120, determine −2

whether the statement is true or false. Justify your answer.
119. If f is an even function, then f ⫺1 exists.
120. If the inverse function of f exists, and the graph of f has 131. Proof Prove that if f and g are one-to-one functions,
a y-intercept, then the y-intercept of f is an x-intercept then f ⬚ g⫺1x ⫽ g⫺1 ⬚ f ⫺1x.
of f ⫺1. 132. Proof Prove that if f is a one-to-one odd function,
then f ⫺1 is an odd function.
Think About It In Exercises 121–124, determine
whether the situation could be represented by a Cumulative Mixed Review
one-to-one function. If so, write a statement that
Simplifying a Rational Expression In Exercises 133–136,
describes the inverse function.
write the rational expression in simplest form.
121. The number of miles n a marathon runner has completed
27x3 5x2y
in terms of the time t in hours 133. 134.
3x2 xy ⫹ 5x
122. The population p of a town in terms of the year t from
1990 through 2010 given that the population was x2 ⫺ 36 x2 ⫹ 3x ⫺ 40
135. 136.
greatest in 2000 6⫺x x2 ⫺ 3x ⫺ 10
123. The depth of the tide d at a beach in terms of the time
Testing for Functions In Exercises 137–140, determine
t over a 24-hour period
whether the equation represents y as a function of x.
124. The height h in inches of a human child from age 2 to
age 14 in terms of his or her age n in years 137. x ⫽ 5 138. y ⫽ x ⫹ 2
139. x2 ⫹ y2 ⫽ 9 140. x ⫺ y2 ⫽ 0
125. Writing Describe the relationship between the graph
of a function f and the graph of its inverse function f ⫺1.

1.7 Linear Models and Scatter Plots

Scatter Plots and Correlation What you should learn

● Construct scatter plots and
Many real-life situations involve finding relationships between two variables, such as
interpret correlation.
the year and the number of employees in the cellular telecommunications industry. In
● Use scatter plots and a graphing
a typical situation, data are collected and written as a set of ordered pairs. The graph
utility to find linear models for
of such a set is called a scatter plot. (For a brief discussion of scatter plots, see
data.
Appendix B.1.)
Why you should learn it
Example 1 Constructing a Scatter Plot Real-life data often follow a linear
pattern. For instance, in Exercise 25
The data in the table show the numbers E (in thousands) of employees in the cellular on page 79, you will find a linear
telecommunications industry in the United States from 2002 through 2007. Construct a model for the winning times in
scatter plot of the data. (Source: CTIA–The Wireless Association) the women’s 400-meter freestyle
Olympic swimming event.

Employees in the cellular

Year telecommunications
industry, E (in thousands)
2002 192
2003 206
2004 226
2005 233
2006 254
2007 267

Solution
Begin by representing the data with a set Cellular Telecommunications
of ordered pairs. Let t represent the year, Industry
with t ⫽ 2 corresponding to 2002. E

共2, 192兲, 共3, 206兲, 共4, 226兲, 280

Number of employees

240
共5, 233兲, 共6, 254兲, 共7, 267兲
(in thousands)

200
Then plot each point in a coordinate plane, 160
as shown in Figure 1.66.
120
80
40
t
2 3 4 5 6 7
Year (2 ↔ 2002)
Now try Exercise 5. Figure 1.66

From the scatter plot in Figure 1.66, it appears that the points describe a
relationship that is nearly linear. The relationship is not exactly linear because the
number of employees did not increase by precisely the same amount each year.
A mathematical equation that approximates the relationship between t and E is a
mathematical model. When developing a mathematical model to describe a set of data,
you strive for two (often conflicting) goals—accuracy and simplicity. For the data
above, a linear model of the form
E ⫽ at ⫹ b
(where a and b are constants) appears to be best. It is simple and relatively accurate.
Patrick Hermans 2010/used under license from Shutterstock.com

Consider a collection of ordered pairs of the form 共x, y兲. If y tends to increase as x
increases, then the collection is said to have a positive correlation. If y tends to
decrease as x increases, then the collection is said to have a negative correlation.
Figure 1.67 shows three examples: one with a positive correlation, one with a negative
correlation, and one with no (discernible) correlation.
y y y

x x x

Positive Correlation Negative Correlation No Correlation

Figure 1.67

Example 2 Interpreting Correlation

On a Friday, 22 students in a class were asked to record the numbers of hours they spent
studying for a test on Monday and the numbers of hours they spent watching television.
The results are shown below. (The first coordinate is the number of hours and the
second coordinate is the score obtained on the test.)
Study Hours: 共0, 40兲, 共1, 41兲, 共2, 51兲, 共3, 58兲, 共3, 49兲, 共4, 48兲, 共4, 64兲,
共5, 55兲, 共5, 69兲, 共5, 58兲, 共5, 75兲, 共6, 68兲, 共6, 63兲, 共6, 93兲, 共7, 84兲, 共7, 67兲,
共8, 90兲, 共8, 76兲, 共9, 95兲, 共9, 72兲, 共9, 85兲, 共10, 98兲
TV Hours: 共0, 98兲, 共1, 85兲, 共2, 72兲, 共2, 90兲, 共3, 67兲, 共3, 93兲, 共3, 95兲, 共4, 68兲,
共4, 84兲, 共5, 76兲, 共7, 75兲, 共7, 58兲, 共9, 63兲, 共9, 69兲, 共11, 55兲, 共12, 58兲, 共14, 64兲,
共16, 48兲, 共17, 51兲, 共18, 41兲, 共19, 49兲, 共20, 40兲
a. Construct a scatter plot for each set of data.
b. Determine whether the points are positively correlated, are negatively correlated, or
have no discernible correlation. What can you conclude?

Solution
a. Scatter plots for the two sets of data are shown in Figure 1.68.
b. The scatter plot relating study hours and test scores has a positive correlation. This
means that the more a student studied, the higher his or her score tended to be. The
scatter plot relating television hours and test scores has a negative correlation. This
means that the more time a student spent watching television, the lower his or her
score tended to be.
y y

100 100

80 80
Test scores

Test scores

60 60

40 40

20 20

x x
2 4 6 8 10 4 8 12 16 20
Study hours TV hours
Figure 1.68
Now try Exercise 7.

Fitting a Line to Data

Finding a linear model to represent the relationship described by a scatter plot is called
fitting a line to data. You can do this graphically by simply sketching the line that
appears to fit the points, finding two points on the line, and then finding the equation of
the line that passes through the two points.

Example 3 Fitting a Line to Data

Find a linear model that relates the year to the number of employees in the cellular
telecommunications industry in the United States. (See Example 1.)

Employees in the cellular Cellular Telecommunications

Year telecommunications Industry
industry, E (in thousands) E
2002 192
280
2003 206

Number of employees
240
2004 226

(in thousands)
200
2005 233
2006 254 160
E = 16t + 158
2007 267 120
80
Solution 40
Let t represent the year, with t ⫽ 2 corresponding to 2002. After plotting the data in the t
table, draw the line that you think best represents the data, as shown in Figure 1.69. Two 2 3 4 5 6 7

points that lie on this line are Year (2 ↔ 2002)

Figure 1.69
共3, 206兲 and 共6, 254兲.
Using the point-slope form, you can find the equation of the line to be
E ⫽ 16共t ⫺ 3兲 ⫹ 206
⫽ 16t ⫹ 158. Linear model

Now try Exercise 11.

Once you have found a model, you can measure how well the model fits the data
by comparing the actual values with the values given by the model, as shown in the
following table. Study Tip
The model in Example 3
t 2 3 4 5 6 7 is based on the two
data points chosen.
Actual E 192 206 226 233 254 267 When different points are
Model E 190 206 222 238 254 270 chosen, the model may change
somewhat. For instance, when
you choose 共5, 233兲 and
The sum of the squares of the differences between the actual values and the model 共7, 267兲, the new model is
values is called the sum of the squared differences. The model that has the least sum
E ⫽ 17共t ⫺ 5) ⫹ 233
is called the least squares regression line for the data. For the model in Example 3, the
sum of the squared differences is 54. The least squares regression line for the data is ⫽ 17t ⫹ 148.
E ⫽ 15.0t ⫹ 162. Best-fitting linear model

Its sum of squared differences is 37. For more on the least squares regression line, see
Appendix C.2 at this textbook’s Companion Website.

Another way to find a linear model to represent the relationship described by

a scatter plot is to enter the data points into a graphing utility and use the linear
regression feature. This method is demonstrated in Example 4.

Technology Tip
For instructions on how to use the linear regression feature, see
Appendix A; for specific keystrokes, go to this textbook’s Companion
Website.

Example 4 A Mathematical Model

The data in the table show the estimated numbers V (in thousands) of alternative-fueled
vehicles in use in the United States from 2001 through 2007. (Source: Energy
Information Administration)

Alternative-fueled vehicles
Year
in use, V (in thousands)
2001 425
2002 471
2003 534
2004 565
2005 592
2006 635
2007 696

a. Use the regression feature of a graphing utility to find a linear model for the data.
Let t represent the year, with t ⫽ 1 corresponding to 2001.
b. How closely does the model represent the data?

Graphical Solution Numerical Solution

a. Use the linear regression feature of a graphing utility to a. Using the linear regression feature of a graphing utility, you
obtain the model shown in Figure 1.70. can find that a linear model for the data is V ⫽ 42.8t ⫹ 388.
b. You can see how well the model fits the data by comparing
the actual values of V with the values of V given by the
model, which are labeled V* in the table below. From the
table, you can see that the model appears to be a good fit for
the actual data.

Figure 1.70
Year V V*
You can approximate the model to be V ⫽ 42.8t ⫹ 388.
b. Graph the actual data and the model. From Figure 1.71, 2001 425 431
it appears that the model is a good fit for the actual data. 2002 471 474
1000
2003 534 516
2004 565 559
2005 592 602
V = 42.8t + 388
0 10 2006 635 645
0
Figure 1.71
2007 696 688

Now try Exercise 15.

Alexey Dudoladov/iStockphoto.com

When you use the regression feature of a graphing calculator or computer program
to find a linear model for data, you will notice that the program may also output an
“r-value.” For instance, the r-value from Example 4 was r ⬇ 0.994. This r-value is the
correlation coefficient of the data and gives a measure of how well the model fits the
data. The correlation coefficient r varies between ⫺1 and 1. Basically, the closer r is ⱍⱍ
to 1, the better the points can be described by a line. Three examples are shown in
Figure 1.72.

18 18 18

0 9 0 9 0 9
0 0 0

r ⫽ 0.972 r ⫽ ⫺0.856 r ⫽ 0.190

Figure 1.72

Technology Tip
For some calculators, the diagnostics on feature must be selected before
the regression feature is used in order to see the value of the correlation
coefficient r. To learn how to use this feature, consult your user’s manual.

Example 5 Finding a Least Squares Regression Line

The following ordered pairs 共w, h兲 represent the shoe sizes w and the heights h (in inches)
of 25 men. Use the regression feature of a graphing utility to find the least squares
regression line for the data.
共10.0, 70.5兲共10.5, 71.0兲共9.5, 69.0兲共11.0, 72.0兲共12.0, 74.0兲
共8.5, 67.0兲共9.0, 68.5兲共13.0, 76.0兲共10.5, 71.5兲共10.5, 70.5兲
共10.0, 71.0兲共9.5, 70.0兲共10.0, 71.0兲共10.5, 71.0兲共11.0, 71.5兲
共12.0, 73.5兲共12.5, 75.0兲共11.0, 72.0兲共9.0, 68.0兲共10.0, 70.0兲
共13.0, 75.5兲共10.5, 72.0兲共10.5, 71.0兲共11.0, 73.0兲共8.5, 67.5兲
Solution
After entering the data into a graphing utility (see Figure 1.73), you obtain the model
shown in Figure 1.74. So, the least squares regression line for the data is
h ⫽ 1.84w ⫹ 51.9.
In Figure 1.75, this line is plotted with the data. Note that the plot does not have
25 points because some of the ordered pairs graph as the same point. The correlation
coefficient for this model is r ⬇ 0.981, which implies that the model is a good fit for
the data.
90

h = 1.84w + 51.9

8 14
50

Figure 1.73 Figure 1.74 Figure 1.75

Now try Exercise 25.

Sean Nel 2010/used under license from Shutterstock.com

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1.7 Exercises For instructions on how to use a graphing utility, see Appendix A.

Vocabulary and Concept Check

In Exercises 1 and 2, fill in the blank.
1. Consider a collection of ordered pairs of the form 共x, y兲. If y tends to increase as x
increases, then the collection is said to have a _______ correlation.
2. To find the least squares regression line for data, you can use the _______ feature
of a graphing utility.

3. In a collection of ordered pairs 共x, y兲, y tends to decrease as x increases. Does the
collection have a positive correlation or a negative correlation?
4. You find the least squares regression line for a set of data. The correlation coefficient
is 0.114. Is the model a good fit?

Procedures and Problem Solving

5. Constructing a Scatter Plot The following ordered 9. y 10. y

pairs give the years of experience x for 15 sales

representatives and the monthly sales y (in thousands of
dollars).
共1.5, 41.7兲, 共1.0, 32.4兲, 共0.3, 19.2兲, 共3.0, 48.4兲,
共4.0, 51.2兲, 共0.5, 28.5兲, 共2.5, 50.4兲, 共1.8, 35.5兲,
共2.0, 36.0兲, 共1.5, 40.0兲, 共3.5, 50.3兲, 共4.0, 55.2兲, x x

共0.5, 29.1兲, 共2.2, 43.2兲, 共2.0, 41.6兲

(a) Create a scatter plot of the data. Fitting a Line to Data In Exercises 11–14, (a) create a
scatter plot of the data, (b) draw a line of best fit that
(b) Does the relationship between x and y appear to be
passes through two of the points, and (c) use the two
approximately linear? Explain.
points to find an equation of the line.
6. Constructing a Scatter Plot The following ordered
pairs give the scores on two consecutive 15-point quizzes 11. 共⫺3, ⫺3兲, 共3, 4兲, 共1, 1兲, 共3, 2兲, 共4, 4兲, 共⫺1, ⫺1兲
for a class of 18 students. 12. 共⫺2, 3兲, 共⫺2, 4兲, 共⫺1, 2兲, 共1, ⫺2兲, 共0, 0兲, 共0, 1兲
共7, 13兲, 共9, 7兲, 共14, 14兲, 共15, 15兲, 共10, 15兲, 共9, 7兲, 13. 共0, 2兲, 共⫺2, 1兲, 共3, 3兲, 共1, 3兲, 共4, 4兲
共14, 11兲, 共14, 15兲, 共8, 10兲, 共9, 10兲, 共15, 9兲, 共10, 11兲, 14. 共3, 2兲, 共2, 3兲, 共1, 5兲, 共4, 0兲, 共5, 0兲
共11, 14兲, 共7, 14兲, 共11, 10兲, 共14, 11兲, 共10, 15兲, 共9, 6兲
A Mathematical Model In Exercises 15 and 16, use the
(a) Create a scatter plot of the data.
regression feature of a graphing utility to find a linear
(b) Does the relationship between consecutive quiz model for the data. Then use the graphing utility to
scores appear to be approximately linear? If not, give decide how closely the model fits the data (a) graphically
some possible explanations. and (b) numerically. To print an enlarged copy of the
graph, go to the website www.mathgraphs.com.
Interpreting Correlation In Exercises 7–10, the scatter
15. y 16. y
plot of a set of data is shown. Determine whether the
points are positively correlated, are negatively correlated, 4 (2, 3) (4, 3) 6
(− 2, 6)
or have no discernible correlation. (− 1, 1) 2 (0, 2) (−1, 4) (1, 1)
7. y 8. y x (0, 2) (2, 1)
(− 3, 0) 2 4
−2 x
−2 2 4
−4 −2

x x

17. MODELING DATA 19. MODELING DATA

Hooke’s Law states that the force F required to The total player payrolls T (in millions of dollars) for
compress or stretch a spring (within its elastic limits) the Pittsburgh Steelers from 2004 through 2008 are
is proportional to the distance d that the spring is shown in the table. (Source: USA Today)
compressed or stretched from its original length. That
is, F ⫽ kd, where k is the measure of the stiffness of Total player payroll, T
the spring and is called the spring constant. The table Year
(in millions of dollars)
shows the elongation d in centimeters of a spring when 2004 78.0
a force of F kilograms is applied.
2005 84.2
2006 94.0
Force, F Elongation, d
2007 106.3
20 1.4 2008 128.8
40 2.5
60 4.0 (a) Use a graphing utility to create a scatter plot of the
80 5.3 data, with t ⫽ 4 corresponding to 2004.
100 6.6 (b) Use the regression feature of the graphing utility to
find a linear model for the data.
(a) Sketch a scatter plot of the data. (c) Use the graphing utility to plot the data and graph
the model in the same viewing window. Is the
(b) Find the equation of the line that seems to best fit model a good fit? Explain.
the data.
(d) Use the model to predict the payrolls in 2010 and
(c) Use the regression feature of a graphing utility to 2015. Do the results seem reasonable? Explain.
find a linear model for the data.
(e) What is the slope of your model? What does it tell
(d) Use the model from part (c) to estimate the elongation you about the player payroll?
of the spring when a force of 55 kilograms is applied.

20. MODELING DATA

18. MODELING DATA
The mean salaries S (in thousands of dollars) of public
The numbers of subscribers S (in millions) to wireless school teachers in the United States from 2002 through
networks from 2002 through 2008 are shown in the table. 2008 are shown in the table. (Source: National
(Source: CTIA–The Wireless Association) Education Association)

Subscribers, S Mean salary, S

Year Year
(in millions) (in thousands of dollars)
2002 140.8 2002 44.7
2003 158.7 2003 45.7
2004 182.1 2004 46.6
2005 207.9 2005 47.7
2006 233.0 2006 49.0
2007 255.4
2007 50.8
2008 270.3
2008 52.3
(a) Use a graphing utility to create a scatter plot of the (a) Use a graphing utility to create a scatter plot of the
data, with t ⫽ 2 corresponding to 2002. data, with t ⫽ 2 corresponding to 2002.
(b) Use the regression feature of the graphing utility to (b) Use the regression feature of the graphing utility to
find a linear model for the data. find a linear model for the data.
(c) Use the graphing utility to plot the data and graph (c) Use the graphing utility to plot the data and graph
the model in the same viewing window. Is the the model in the same viewing window. Is the
model a good fit? Explain. model a good fit? Explain.
(d) Use the model to predict the number of subscribers (d) Use the model to predict the mean salaries in 2016
in 2015. Is your answer reasonable? Explain. and 2018. Do the results seem reasonable? Explain.

21. MODELING DATA 23. MODELING DATA

The projected populations P (in thousands) of New The table shows the advertising expenditures x and sales
Jersey for selected years, based on the 2000 census, are volumes y for a company for seven randomly selected
shown in the table. (Source: U.S. Census Bureau) months. Both are measured in thousands of dollars.

Population, P Advertising Sales

Year Month
(in thousands) expenditures, x volume, y
2010 9018 1 2.4 202
2015 9256 2 1.6 184
2020 9462 3 2.0 220
2025 9637 4 2.6 240
2030 9802 5 1.4 180
6 1.6 164
(a) Use a graphing utility to create a scatter plot of the 7 2.0 186
data, with t ⫽ 10 corresponding to 2010.
(b) Use the regression feature of the graphing utility (a) Use the regression feature of a graphing utility to
to find a linear model for the data. find a linear model for the data.
(c) Use the graphing utility to plot the data and graph (b) Use the graphing utility to plot the data and graph
the model in the same viewing window. the model in the same viewing window.
(d) Create a table showing the actual values of P and (c) Interpret the slope of the model in the context of
the values of P given by the model. How closely the problem.
does the model fit the data? (d) Use the model to estimate sales for advertising
(e) Use the model to predict the population of New Jersey expenditures of $1500.
in 2050. Does the result seem reasonable? Explain.

24. MODELING DATA

22. MODELING DATA The table shows the numbers T of stores owned by the
The projected populations P (in thousands) of Wyoming Target Corporation from 2000 through 2008.
for selected years, based on the 2000 census, are shown (Source: Target Corp.)
in the table. (Source: U.S. Census Bureau)
Year Number of stores, T
Population, P
Year 2000 1307
(in thousands)
2001 1381
2010 520
2002 1475
2015 528
2003 1553
2020 531
2004 1308
2025 529
2005 1397
2030 523
2006 1488
(a) Use a graphing utility to create a scatter plot of the 2007 1591
data, with t ⫽ 10 corresponding to 2010. 2008 1682
(b) Use the regression feature of the graphing utility to
find a linear model for the data. (a) Use a graphing utility to make a scatter plot of the
data, with t ⫽ 0 corresponding to 2000. Identify
(c) Use the graphing utility to plot the data and graph two sets of points in the scatter plot that are
the model in the same viewing window. approximately linear.
(d) Create a table showing the actual values of P and (b) Use the regression feature of the graphing utility to
the values of P given by the model. How closely find a linear model for each set of points.
does the model fit the data?
(c) Write a piecewise-defined model for the data. Use the
(e) Use the model to predict the population of Wyoming graphing utility to graph the piecewise-defined model.
in 2050. Does the result seem reasonable? Explain.
(d) Describe a scenario that could be the cause of the
break in the data.

25. (p. 71) The following 28. When the correlation coefficient for a linear regression
ordered pairs 共t, T兲 represent the Olympic model is close to ⫺1, the regression line is a poor fit for
year t and the winning time T (in minutes) the data.
in the women’s 400-meter freestyle swimming
event. (Source: International Olympic 29. Writing Use your school’s library, the Internet, or
Committee) some other reference source to locate data that you
think describes a linear relationship. Create a scatter
共1952, 5.20兲共1972, 4.32兲共1992, 4.12兲 plot of the data and find the least squares regression line
共1956, 4.91兲共1976, 4.16兲共1996, 4.12兲 that represents the points. Interpret the slope and
共1960, 4.84兲共1980, 4.15兲共2000, 4.10兲 y-intercept in the context of the data. Write a summary
of your findings.
共1964, 4.72兲共1984, 4.12兲共2004, 4.09兲
共1968, 4.53兲共1988, 4.06兲共2008, 4.05兲 30. CAPSTONE Each graphing utility screen below shows
(a) Use the regression feature of a graphing utility to the least squares regression line for a set of data. The
find a linear model for the data and to identify the equations and r-values for the models are given.
correlation coefficient. Let t represent the year, with y ⫽ 0.68x ⫹ 2.7 (i) 12
t ⫽ 0 corresponding to 1950. y ⫽ 0.41x ⫹ 2.7
(b) What information is given by the sign of the slope y ⫽ ⫺0.62x ⫹ 10.0
of the model?
r ⫽ 0.973
(c) Use the graphing utility to plot the data and graph r ⫽ ⫺0.986 0 9
0
the model in the same viewing window. r ⫽ 0.624
(d) Create a table showing the actual values of y and the
(ii) 12 (iii) 12
values of y given by the model. How closely does
the model fit the data?
(e) How can you use the value of the correlation
coefficient to help answer the question in part (d)?
0 9 0 9
(f) Would you use the model to predict the winning 0 0
times in the future? Explain.
(a) Determine the equation and correlation coefficient
26. MODELING DATA (r-value) that represents each graph. Explain how
you found your answers.
In a study, 60 colts were measured every 14 days from
(b) According to the correlation coefficients, which
birth. The ordered pairs 共d, l兲 represent the average
model is the best fit for its data? Explain.
length l (in centimeters) of the 60 colts d days after
birth: 共14, 81.2兲, 共28, 87.1兲, 共42, 93.7兲, 共56, 98.3兲,
共70, 102.4兲, 共84, 106.2兲, and 共98, 110.0兲. (Source:
American Society of Animal Science)
Cumulative Mixed Review
(a) Use the regression feature of a graphing utility to Evaluating a Function In Exercises 31 and 32, evaluate
find a linear model for the data and to identify the the function at each value of the independent variable
correlation coefficient. and simplify.
(b) According to the correlation coefficient, does the 31. f 共x兲 ⫽ 2x2 ⫺ 3x ⫹ 5
model represent the data well? Explain. (a) f 共⫺1兲
(c) Use the graphing utility to plot the data and graph (b) f 共w ⫹ 2兲
the model in the same viewing window. How closely
32. g共x兲 ⫽ 5x2 ⫺ 6x ⫹ 1
does the model fit the data?
(a) g共⫺2兲
(d) Use the model to predict the average length of a
colt 112 days after birth. (b) g共z ⫺ 2兲

Solving Equations In Exercises 33–36, solve the equation

Conclusions algebraically. Check your solution graphically.
True or False? In Exercises 27 and 28, determine 33. 6x ⫹ 1 ⫽ ⫺9x ⫺ 8
whether the statement is true or false. Justify your answer. 34. 3共x ⫺ 3兲 ⫽ 7x ⫹ 2
27. A linear regression model with a positive correlation 35. 8x2 ⫺ 10x ⫺ 3 ⫽ 0
will have a slope that is greater than 0. 36. 10x2 ⫺ 23x ⫺ 5 ⫽ 0
Patrick Hermans 2010/used under license from Shutterstock.com

1 Chapter Summary

What did you learn? Explanation and Examples Review

Exercises
Find the slopes of lines (p. 3). The slope m of the nonvertical line through 共x1, y1兲 and 共x2, y2兲 is
y ⫺ y1 1–8
m⫽ 2 , x ⫽ x2.
x2 ⫺ x1 1

Write linear equations given points The point-slope form of the equation of the line that passes
on lines and their slopes (p. 5). through the point 共x1, y1兲 and has a slope of m is 9–16
1.1 y ⫺ y1 ⫽ m共x ⫺ x1兲.

Use slope-intercept forms of linear The graph of the equation y ⫽ mx ⫹ b is a line whose slope is m
equations to sketch lines (p. 7). and whose y-intercept is 共0, b兲. 17–30

Use slope to identify parallel and Parallel lines: Slopes are equal.
perpendicular lines (p. 9). 31, 32
Perpendicular lines: Slopes are negative reciprocals of each other.

Decide whether a relation between A function f from a set A to a set B is a relation that assigns to
two variables represents a function each element x in the set A exactly one element y in the set B.
(p. 16). The set A is the domain (or set of inputs) of the function f, and 33–42
the set B contains the range (or set of outputs).

Use function notation and evaluate Equation: f 共x兲 ⫽ 5 ⫺ x2

functions (p. 18), and find the f 冇2冈: f 共2兲 ⫽ 5 ⫺ 22 ⫽ 1 43–50
1.2 domains of functions (p. 20).
Domain of f 冇x冈 ⴝ 5 ⴚ x2: All real numbers x

Use functions to model and solve A function can be used to model the number of construction
real-life problems (p. 22). employees in the United States. (See Example 8.) 51, 52

Evaluate difference quotients (p. 23). f 共x ⫹ h兲 ⫺ f 共x兲

Difference quotient: ,h⫽0 53, 54
h
y
Find the domains and ranges of
(2, 4)
functions (p. 29). 4
3 y = f(x)
2
1 (4, 0)
x
−3 − 2 − 1 1 2 3 4 5 6 55–62
Range

(− 1, − 5)
Domain

Use the Vertical Line Test for A set of points in a coordinate plane is the graph of y as a function
1.3 functions (p. 30). of x if and only if no vertical line intersects the graph at more 63–66
than one point.

Determine intervals on which A function f is increasing on an interval when, for any x1 and x2 in
functions are increasing, the interval,
decreasing, or constant (p. 31). x1 < x2 implies f 共x1兲 < f 共x2兲.
A function f is decreasing on an interval when, for any x1 and x2
in the interval, 67–70
x1 < x2 implies f 共x1兲 > f 共x2兲.
A function f is constant on an interval when, for any x1 and x2 in
the interval,
f 共x1兲 ⫽ f 共x2兲.

What did you learn? Explanation and Examples Review

Exercises
Determine relative maximum and A function value f 共a兲 is called a relative minimum of f
relative minimum values of when there exists an interval 共x1, x2兲 that contains a such
functions (p. 32). that x1 < x < x2 implies f 共a兲 ⱕ f 共x兲. A function value f 共a兲
71–74
is called a relative maximum of f when there exists an
interval 共x1, x2兲 that contains a such that x1 < x < x2
implies f 共a兲 ⱖ f 共x兲.
1.3
Identify and graph step functions Greatest integer: f 共x兲 ⫽ 冀x冁
and other piecewise-defined 75–78
functions (p. 34).

Identify even and odd functions Even: For each x in the domain of f, f 共⫺x兲 ⫽ f 共x兲.
(p. 35). 79–86
Odd: For each x in the domain of f, f 共⫺x兲 ⫽ ⫺f 共x兲.

Recognize graphs of parent Linear: f 共x兲 ⫽ x; Quadratic: f 共x兲 ⫽ x2;

functions (p. 41). Cubic: f 共x兲 ⫽ x3; Absolute value: f 共x兲 ⫽ x ; ⱍⱍ 87–92
Square root: f 共x兲 ⫽ 冪x; Rational: f 共x兲 ⫽ 1兾x
(See Figure 1.34, page 41.)

1.4 Use vertical and horizontal shifts Vertical shifts: h共x兲 ⫽ f 共x兲 ⫹ c or h共x兲 ⫽ f 共x兲 ⫺ c
(p. 42), reflections (p. 44), and Horizontal shifts: h共x兲 ⫽ f 共x ⫺ c兲 or h共x兲 ⫽ f 共x ⫹ c兲
nonrigid transformations (p. 46)
to graph functions. Reflection in the x-axis: h共x兲 ⫽ ⫺f 共x兲 93–106
Reflection in the y-axis: h共x兲 ⫽ f 共⫺x兲
Nonrigid transformations: h共x兲 ⫽ cf 共x兲 or h共x兲 ⫽ f 共cx兲

Add, subtract, multiply, and 共 f ⫹ g兲共x兲 ⫽ f 共x兲 ⫹ g共x兲共 f ⫺ g兲共x兲 ⫽ f 共x兲 ⫺ g共x兲
divide functions (p. 50), find the 共 fg兲共x兲 ⫽ f 共x兲 ⭈ g共x兲共 f兾g兲共x兲 ⫽ f 共x兲兾g共x兲, g共x兲 ⫽ 0
compositions of functions (p. 52), 107–122
and write a function as a composition Composition of functions: 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲
of two functions (p. 54).
1.5
Use combinations of functions to A composite function can be used to represent the number
model and solve real-life problems of bacteria in a petri dish as a function of the amount of
123, 124
(p. 55). time the petri dish has been out of refrigeration. (See
Example 10.)

Find inverse functions informally Let f and g be two functions such that f 共 g共x兲兲 ⫽ x for every
and verify that two functions are x in the domain of g and g共 f 共x兲兲 ⫽ x for every x in the
125–128
inverse functions of each other domain of f. Under these conditions, the function g is the
(p. 60). inverse function of the function f.

Use graphs of functions to decide If the point 共a, b兲 lies on the graph of f, then the point 共b, a兲
whether functions have inverse must lie on the graph of f ⫺1, and vice versa. In short, f ⫺1 is 129, 130
1.6 functions (p. 63). a reflection of f in the line y ⫽ x.

Determine whether functions are A function f is one-to-one when, for a and b in its domain,
131–134
one-to-one (p. 64). f 共a兲 ⫽ f 共b兲 implies a ⫽ b.

Find inverse functions To find inverse functions, replace f 共x兲 by y, interchange the
135–142
algebraically (p. 65). roles of x and y, and solve for y. Replace y by f ⫺1共x兲.

Construct scatter plots (p. 71) and A scatter plot is a graphical representation of data written as
143–146
interpret correlation (p. 72). a set of ordered pairs.
1.7 Use scatter plots (p. 73) and a The best-fitting linear model can be found using the linear
graphing utility (p. 74) to find regression feature of a graphing utility or a computer 147, 148
linear models for data. program.

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1 Review Exercises For instructions on how to use a graphing utility, see Appendix A.

1.1 29. Business During the second and third quarters of the
year, an e-commerce business had sales of $160,000
Finding the Slope of a Line In Exercises 1–8, plot the
and $185,000, respectively. The growth of sales follows
two points and find the slope of the line passing through
a linear pattern. Estimate sales during the fourth quarter.
the points.
30. Accounting The dollar value of a DVD player in
1. 共⫺3, 2兲, 共8, 2兲 2. 共3, ⫺1兲, 共⫺3, ⫺1兲 2010 is $134. The product will decrease in value at an
3. 共7, ⫺1兲, 共7, 12兲 4. 共8, ⫺1兲, 共8, 2兲 expected rate of $26.80 per year.
5. 共 2 , 1兲, 共5, 2 兲共⫺ 34, 56 兲, 共12, ⫺ 52 兲
3 5
6. (a) Write a linear equation that gives the dollar value V
7. 共⫺4.5, 6兲, 共2.1, 3兲 8. 共⫺2.7, ⫺6.3兲, 共0, 1.8兲 of the DVD player in terms of the year t. (Let t ⫽ 0
represent 2010.)
The Point-Slope Form of the Equation of a Line In (b) Use a graphing utility to graph the equation found
Exercises 9–16, (a) use the point on the line and the slope in part (a). Be sure to choose an appropriate viewing
of the line to find an equation of the line, and (b) find window. State the dimensions of your viewing
three additional points through which the line passes. window, and explain why you chose the values that
(There are many correct answers.) you did.
Point Slope (c) Use the value or trace feature of the graphing utility
to estimate the dollar value of the DVD player
9. 共2, ⫺1兲 m ⫽ 14
in 2014. Confirm your answer algebraically.
10. 共⫺3, 5兲 m ⫽ ⫺ 32
(d) According to the model, when will the DVD player
11. 共0, ⫺5兲 m ⫽ 32 have no value?
12. 共0, 78 兲 m ⫽ ⫺ 45
13. 共⫺2, 6兲 m⫽0 Equations of Parallel and Perpendicular Lines In
Exercises 31 and 32, write the slope-intercept forms
14. 共⫺8, 8兲 m⫽0
of the equations of the lines through the given point
15. 共10, ⫺6兲 m is undefined. (a) parallel to the given line and (b) perpendicular to the
16. 共5, 4兲 m is undefined. given line. Verify your result with a graphing utility (use
a square setting).
Finding the Slope-Intercept Form In Exercises 17–24,
Point Line
write an equation of the line that passes through the
points. Use the slope-intercept form, if possible. If not 31. 共3, ⫺2兲 5x ⫺ 4y ⫽ 8
possible, explain why. Use a graphing utility to graph the 32. 共⫺8, 3兲 2x ⫹ 3y ⫽ 5
line (if possible).
1.2
17. 共2, ⫺1兲, 共4, ⫺1兲 18. 共0, 0兲, 共0, 10兲 Testing for Functions In Exercises 33 and 34, which set of
19. 共7, 113 兲, 共9, 113 兲 20. 共8, 4兲, 共8, ⫺6兲
5 5
ordered pairs represents a function from A to B? Explain.
21. 共⫺1, 0兲, 共6, 2兲 22. 共1, 6兲, 共4, 2兲 33. A ⫽ 再10, 20, 30, 40冎 and B ⫽ 再0, 2, 4, 6冎
共3, ⫺1兲, 共⫺3, 2兲 24. 共⫺ 2, 1兲, 共⫺4, 9 兲
5 2
23. (a) 再共20, 4兲, 共40, 0兲, 共20, 6兲, 共30, 2兲冎
Using a Rate of Change to Write an Equation In (b) 再共10, 4兲, 共20, 4兲, 共30, 4兲, 共40, 4兲冎
Exercises 25–28, you are given the dollar value of a product 34. A ⫽ 再u, v, w冎 and B ⫽ 再⫺2, ⫺1, 0, 1, 2冎
in 2010 and the rate at which the value of the item is (a) 再共u, ⫺2兲, 共v, 2兲, 共w, 1兲冎
expected to change during the next 5 years. Use this (b) 再共w, ⫺2兲, 共v, 0兲, 共w, 2兲冎
information to write a linear equation that gives the dollar
value V of the product in terms of the year t. (Let t ⴝ 0 Testing for Functions Represented Algebraically In
represent 2010.) Exercises 35– 42, determine whether the equation
2010 Value Rate represents y as a function of x.
25. $12,500 $850 increase per year 35. 16x 2 ⫺ y 2 ⫽ 0 36. x3 ⫹ y2 ⫽ 64
26. $3795 $115 decrease per year 37. 2x ⫺ y ⫺ 3 ⫽ 0 38. 2x ⫹ y ⫽ 10
27. $625.50 $42.70 increase per year 39. y ⫽ 冪1 ⫺ x 40. y ⫽ 冪x2 ⫹ 4
28. $72.95 $5.15 decrease per year 41. ⱍⱍ
y ⫽x⫹2 42. ⱍⱍ
16 ⫺ y ⫺ 4x ⫽ 0

Evaluating a Function In Exercises 43–46, evaluate the 1.3

function at each specified value of the independent
Finding the Domain and Range of a Function In
variable, and simplify.
Exercises 55–62, use a graphing utility to graph the
43. f 共x兲 ⫽ x2 ⫹ 1 function and estimate its domain and range. Then find
(a) f 共1兲 (b) f 共⫺3兲 the domain and range algebraically.
(c) f 共b3兲 (d) f 共x ⫺ 1兲 55. f 共x兲 ⫽ 3 ⫺ 2x2 56. f 共x兲 ⫽ 2x2 ⫹ 5
44. g共x兲 ⫽ 冪x ⫹ 12 57. f 共x兲 ⫽ 冪x ⫹ 3 ⫹ 4 58. f 共x兲 ⫽ 2 ⫺ 冪x ⫺ 5
(a) g共⫺1兲 (b) g共3兲 59. h 共x兲 ⫽ 冪36 ⫺ x2 60. f 共x兲 ⫽ 冪x2 ⫺ 9
(c) g共3x兲 (d) g共x ⫹ 2兲 61. ⱍ
f 共x兲 ⫽ x ⫹ 5 ⫹ 2 ⱍ 62. ⱍ
f 共x兲 ⫽ x ⫹ 1 ⫺ 3 ⱍ
45. h共x兲 ⫽ 2 冦
2x ⫹ 1, x ⱕ ⫺1
x ⫹ 2, x > ⫺1 Vertical Line Test for Functions In Exercises 63–66, use
the Vertical Line Test to determine whether y is a function
(a) h共⫺2兲 (b) h共⫺1兲 of x. Describe how to enter the equation into a graphing
(c) h共0兲 (d) h共2兲 utility to produce the given graph.

46. f 共x兲 ⫽
3
2x ⫺ 5
63. y ⫺ 4x ⫽ x2 ⱍ
64. x ⫹ 5 ⫺ 2y ⫽ 0 ⱍ
3 7
(a) f 共1兲 (b) f 共⫺2兲
(c) f 共t兲 (d) f 共10兲 −8 4

−10 2
Finding the Domain of a Function In Exercises 47–50,
find the domain of the function. −5 −1

x⫺1 x2 65. 3x ⫹ y2 ⫺ 2 ⫽ 0 66. x2 ⫹ y2 ⫺ 49 ⫽ 0

47. f 共x兲 ⫽ 48. f 共x兲 ⫽
x⫹2 x ⫹1
2 8 8

49. f 共x兲 ⫽ 冪25 ⫺ x 2 50. f 共x兲 ⫽ 冪x 2 ⫺ 16

−22 2 −12 12
51. Industrial Engineering A hand tool manufacturer
produces a product for which the variable cost is $5.35
per unit and the fixed costs are $16,000. The company −8 −8

sells the product for $8.20 and can sell all that it
produces. Increasing and Decreasing Functions In Exercises
(a) Write the total cost C as a function of x, the number 67–70, (a) use a graphing utility to graph the function
of units produced. and (b) determine the open intervals on which the function
is increasing, decreasing, or constant.
(b) Write the profit P as a function of x.
52. Education The numbers n (in millions) of students 67. f 共x兲 ⫽ x3 ⫺ 3x 68. f 共x兲 ⫽ 冪x2 ⫺ 9
enrolled in public schools in the United States from 69. f 共x兲 ⫽ x冪x ⫺ 6 70. f 共x兲 ⫽
x⫹8 ⱍ ⱍ
2000 through 2008 can be approximated by 2

n共t兲 ⫽ 冦⫺0.3333t
0.76t ⫹ 61.4, 0 ⱕ t ⱕ 4
⫹ 6.6t ⫺ 42.37t ⫹ 152.7,
3 2 4 < tⱕ 8
Approximating Relative Minima and Maxima In
Exercises 71–74, use a graphing utility to approximate
where t is the year, with t ⫽ 0 corresponding to 2000. (to two decimal places) any relative minimum or relative
(Source: U.S. Census Bureau) maximum values of the function.
(a) Use the table feature of a graphing utility to approximate 71. f 共x兲 ⫽ 共x 2 ⫺ 4兲 2 72. f 共x兲 ⫽ x2 ⫺ x ⫺ 1
the enrollment from 2000 through 2008.
73. h共x兲 ⫽ 4x 3 ⫺ x4 74. f 共x兲 ⫽ x3 ⫺ 4x2 ⫺ 1
(b) Use the graphing utility to graph the model and
estimate the enrollment for the years 2009 through Sketching Graphs In Exercises 75–78, sketch the graph
2012. Do the values seem reasonable? Explain. of the function by hand.

冦冦
1
Evaluating a Difference Quotient In Exercises 53 and 3x ⫹ 5, x < 0 x ⫹ 3, x < 0
75. f 共x兲 ⫽ 76. f 共x兲 ⫽ 2
f 冇x ⴙ h冈 ⴚ f 冇x冈 x ⫺ 4, x ⱖ 0 4 ⫺ x2, x ⱖ 0
54, find the difference quotient for the
h 77. f 共x兲 ⫽ 冀x冁 ⫹ 3 78. f 共x兲 ⫽ 冀x ⫹ 2冁
given function and simplify your answer.
53. f 共x兲 ⫽ 2x2 ⫹ 3x ⫺ 1 54. f 共x兲 ⫽ x2 ⫺ 3x ⫹ 5

Even and Odd Functions In Exercises 79–86, determine ⫺2 1

105. h共x兲 ⫽ ⫺3 106. h共x兲 ⫽ ⫺4
algebraically whether the function is even, odd, or neither. x⫹1 x⫹2
Verify your answer using a graphing utility.
1.5
79. f 共x兲 ⫽ x2 ⫹ 6 80. f 共x兲 ⫽ x2 ⫺ x ⫺ 1 Evaluating a Combination of Functions In Exercises
81. f 共x兲 ⫽ 共x2 ⫺ 8兲2 82. f 共x兲 ⫽ 2x3 ⫺ x2 107–116, let f 冇x冈 ⴝ 3 ⴚ 2x, g冇x冈 ⴝ 冪x, and
83. f 共x兲 ⫽ 3x5兾 2 84. f 共x兲 ⫽ 3x 2兾5 h冇x冈 ⴝ 3x2 ⴙ 2, and find the indicated values.
85. f 共x兲 ⫽ 冪4 ⫺ x2 86. f 共x兲 ⫽ x冪x2 ⫺ 1 107. 共 f ⫺ g兲共4兲 108. 共 f ⫹ h兲共5兲
1.4 109. 共 f ⫹ g兲共25兲 110. 共g ⫺ h兲共1兲

Library of Parent Functions In Exercises 87–92,

identify the parent function and describe the
111. 共 fh兲共1兲 112.
g
h
共1兲冢冣
transformation shown in the graph. Write an equation 113. 共h ⬚ g兲共5兲 114. 共g ⬚ f 兲共⫺3兲
for the graphed function. 115. 共 f ⬚ h兲共⫺4兲 116. 共g ⬚ h兲共6兲
87. 4 88. 6
Identifying a Composite Function In Exercises 117–122,
find two functions f and g such that 冇 f ⬚ g冈冇x冈 ⴝ h冇x冈.
−3 9 (There are many correct answers.)
−2 10
117. h共x兲 ⫽ 共x ⫹ 3兲2 118. h共x兲 ⫽ 共1 ⫺ 2x兲3
−4 −2
119. h共x兲 ⫽ 冪4x ⫹ 2 120. h共x兲 ⫽ 冪
3
共x ⫹ 2兲2
89. 10 90. 2
4 6
121. h共x兲 ⫽ 122. h共x兲 ⫽
−6 6 x⫹2 共3x ⫹ 1兲3

−6 12 Education In Exercises 123 and 124, the numbers (in

−2 −6
thousands) of students taking the SAT 冇 y1冈 and the ACT
冇 y2冈 for the years 2000 through 2009 can be modeled by
91. 7 92. 4
y1 ⴝ ⴚ2.61t2 ⴙ 55.0t ⴙ 1244 and

−1 11 y2 ⴝ 0.949t3 ⴚ 8.02t2 ⫹ 44.4t ⫹ 1056

−6 6 where t represents the year, with t ⴝ 0 corresponding to
−1 −4 2000. (Source: College Entrance Examination Board and
ACT, Inc.)
Sketching Transformations In y
123. Use a graphing utility to graph y1, y2, and y1 ⫹ y2 in
Exercises 93–96, use the graph 6 y = f(x)
the same viewing window.
of y ⴝ f 冇x冈 to graph the function. (−1, 2)
(4, 2) 124. Use the model y1 ⫹ y2 to estimate the total number of
93. y ⫽ f 共⫺x兲 x students taking the SAT and the ACT in 2010.
−4 −2 2 4 8
94. y ⫽ ⫺f 共x兲
−4 (8, −4) 1.6
95. y ⫽ f 共x兲 ⫺ 2 (−4, −4) Finding Inverse Functions Informally In Exercises
96. y ⫽ f 共x ⫺ 1兲 125–128, find the inverse function of f informally. Verify
that f 冇 f ⴚ1 冇x冈冈 ⴝ x and f ⴚ1冇 f 冇x冈冈 ⴝ x.
Describing Transformations In Exercises 97–106, h is
related to one of the six parent functions on page 41. 125. f 共x兲 ⫽ 6x 126. f 共x兲 ⫽ x ⫹ 5
(a) Identify the parent function f. (b) Describe the 1 x⫺4
sequence of transformations from f to h. (c) Sketch the 127. f 共x兲 ⫽ x ⫹ 3 128. f 共x兲 ⫽
2 5
graph of h by hand. (d) Use function notation to write h
in terms of the parent function f. Algebraic-Graphical-Numerical In Exercises 129 and
1 1 130, show that f and g are inverse functions (a) algebraically,
97. h共x兲 ⫽ ⫺ 6 98. h共x兲 ⫽ ⫺ ⫺ 3 (b) graphically, and (c) numerically.
x x
99. h 共x兲 ⫽ 共x ⫺ 2兲3 ⫹ 5 100. h共x兲 ⫽ ⫺ 共x ⫺ 2兲2 ⫺ 8 3⫺x
129. f 共x兲 ⫽ 3 ⫺ 4x, g共x兲 ⫽
101. h 共x兲 ⫽ ⫺ 冪x ⫹ 6 102. h共x兲 ⫽ 冪x ⫺ 1 ⫹ 4 4
103. h 共x兲 ⫽ x ⫹ 9 ⱍⱍ 104. h共x兲 ⫽ x ⫹ 8 ⫺ 1 ⱍ ⱍ 130. f 共x兲 ⫽ 冪x ⫹ 1, g共x兲 ⫽ x2 ⫺ 1, x ⱖ 0

Using the Horizontal Line Test In Exercises 131–134, 147. MODELING DATA
use a graphing utility to graph the function and use the
In an experiment, students measured the speed s (in
Horizontal Line Test to determine whether the function
meters per second) of a ball t seconds after it was
is one-to-one and an inverse function exists.
released. The results are shown in the table.
131. f 共x兲 ⫽ 12 x ⫺ 3
132. f 共x兲 ⫽ 共x ⫺ 1兲2 Time, t Speed, s
2 0 0
133. h共t兲 ⫽
t⫺3 1 11.0
134. g共x兲 ⫽ 冪x ⫹ 6 2 19.4
3 29.2
Finding an Inverse Function Algebraically In Exercises 4 39.4
135–142, find the inverse function of f algebraically.
1 7x ⫹ 3 (a) Sketch a scatter plot of the data.
135. f 共x兲 ⫽ x ⫺ 5 136. f 共x兲 ⫽
2 8 (b) Find the equation of the line that seems to fit the
137. f 共x兲 ⫽ 4x3 ⫺ 3 138. f 共x兲 ⫽ 5x3 ⫹ 2 data best.
139. f 共x兲 ⫽ 冪x ⫹ 10 140. f 共x兲 ⫽ 4冪6 ⫺ x (c) Use the regression feature of a graphing utility to
find a linear model for the data and identify the
141. f 共x兲 ⫽ 4x2 ⫹ 1, x ⱖ 0
1
correlation coefficient.
142. f 共x兲 ⫽ 5 ⫺ 19 x2, xⱖ 0 (d) Use the model from part (c) to estimate the speed
1.7
of the ball after 2.5 seconds.
Interpreting Correlation In Exercises 143 and 144, the
scatter plot of a set of data is shown. Determine whether 148. MODELING DATA
the points are positively correlated, are negatively
The following ordered pairs 共x, y兲 represent the
correlated, or have no discernible correlation.
Olympic year x and the winning time y (in minutes) in
143. y 144. y the men’s 400-meter freestyle swimming event.
(Source: International Olympic Committee)
共1964, 4.203兲共1980, 3.855兲共1996, 3.800兲
x x 共1968, 4.150兲共1984, 3.854兲共2000, 3.677兲
共1972, 4.005兲共1988, 3.783兲共2004, 3.718兲
共1976, 3.866兲共1992, 3.750兲共2008, 3.698兲
(a) Use the regression feature of a graphing utility to
145. Education The following ordered pairs give the
find a linear model for the data. Let x represent the
entrance exam scores x and the grade-point averages y
year, with x ⫽ 4 corresponding to 1964.
after 1 year of college for 10 students.
(b) Use the graphing utility to create a scatter plot of the
共75, 2.3兲, 共82, 3.0兲, 共90, 3.6兲, 共65, 2.0兲, 共70, 2.1兲,
data. Graph the model in the same viewing window.
共88, 3.5兲, 共93, 3.9兲, 共69, 2.0兲, 共80, 2.8兲, 共85, 3.3兲
(c) Is the model a good fit for the data? Explain.
(a) Create a scatter plot of the data.
(d) Is this model appropriate for predicting the winning
(b) Does the relationship between x and y appear to be
times in future Olympics? Explain.
approximately linear? Explain.
146. Industrial Engineering A machine part was tested
by bending it x centimeters 10 times per minute until Conclusions
it failed (y equals the time to failure in hours). The
results are given as the following ordered pairs. True or False? In Exercises 149–151, determine whether
the statement is true or false. Justify your answer.
共3, 61兲, 共6, 56兲, 共9, 53兲, 共12, 55兲, 共15, 48兲, 共18, 35兲,
共21, 36兲, 共24, 33兲, 共27, 44兲, 共30, 23兲 149. If the graph of the parent function f 共x兲 ⫽ x2 is moved
(a) Create a scatter plot of the data. six units to the right, moved three units upward, and
reflected in the x-axis, then the point 共⫺1, 28兲 will lie
(b) Does the relationship between x and y appear to be
on the graph of the transformation.
approximately linear? If not, give some possible
explanations. 150. If f 共x兲 ⫽ x n where n is odd, then f ⫺1 exists.
151. There exists no function f such that f ⫽ f ⫺1.

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1 Chapter Test For instructions on how to use a graphing utility, see Appendix A.

Take this test as you would take a test in class. After you are finished, check your
4
y 2(4 − x) = x 3
work against the answers in the back of the book.
1. Find the equations of the lines that pass through the point 共0, 4兲 and are (a) parallel
to and (b) perpendicular to the line 5x ⫹ 2y ⫽ 3. −4 8

2. Find the slope-intercept form of the equation of the line that passes through the
points 共2, ⫺1兲 and 共⫺3, 4兲. −4

3. Does the graph at the right represent y as a function of x? Explain. Figure for 3

ⱍ ⱍ
4. Evaluate f 共x兲 ⫽ x ⫹ 2 ⫺ 15 at each value of the independent variable and
simplify.
(a) f 共⫺8兲 (b) f 共14兲 (c) f 共t ⫺ 6兲
5. Find the domain of f 共x兲 ⫽ 10 ⫺ 冪3 ⫺ x.
6. An electronics company produces a car stereo for which the variable cost is $25.60
per unit and the fixed costs are $24,000. The product sells for $99.50. Write the
total cost C as a function of the number of units produced and sold, x. Write the
profit P as a function of the number of units produced and sold, x.

In Exercises 7 and 8, determine algebraically whether the function is even, odd, or

neither.
7. f 共x兲 ⫽ 2x3 ⫺ 3x 8. f 共x兲 ⫽ 3x4 ⫹ 5x2

In Exercises 9 and 10, determine the open intervals on which the function is
increasing, decreasing, or constant.
9. h共x兲 ⫽ 14x 4 ⫺ 2x 2 ⱍ
10. g共t兲 ⫽ t ⫹ 2 ⫺ t ⫺ 2 ⱍ ⱍ ⱍ
In Exercises 11 and 12, use a graphing utility to graph the functions and to
approximate (to two decimal places) any relative minimum or relative maximum
values of the function.
11. f 共x兲 ⫽ ⫺x3 ⫺ 5x2 ⫹ 12 12. f 共x兲 ⫽ x5 ⫺ x3 ⫹ 2

In Exercises 13–15, (a) identify the parent function f, (b) describe the sequence of
transformations from f to g, and (c) sketch the graph of g.
13. g共x兲 ⫽ ⫺2共x ⫺ 5兲3 ⫹ 3 14. g共x兲 ⫽ 冪⫺x ⫺ 7 15. g 共x兲 ⫽ 4 ⫺x ⫺ 7 ⱍ ⱍ
16. Use the functions f 共x兲 ⫽ x 2 and g共x兲 ⫽ 冪2 ⫺ x to find the specified function and
its domain.
Average
冢冣
f
(a) 共 f ⫺ g兲共x兲 (b) 共x兲 (c) 共 f ⬚ g兲共x兲 (d) 共g ⬚ f 兲共x兲 Year, t monthly
g cost, C
(in dollars)
In Exercises 17–19, determine whether the function has an inverse function, and if 0 30.37
so, find the inverse function. 1 32.87
3x冪x 2 34.71
17. f 共x兲 ⫽ x3 ⫹ 8 18. f 共x兲 ⫽ x2 ⫹ 6 19. f 共x兲 ⫽
8 3 36.59
4 38.14
20. The table shows the average monthly cost C of basic cable television from 2000 5 39.63
through 2008, where t represents the year, with t ⫽ 0 corresponding to 2000. Use
6 41.17
the regression feature of a graphing utility to find a linear model for the data. Use
7 42.72
the model to estimate the year in which the average monthly cost reached $50.
(Source: SNL Kagan) 8 44.28

Proofs in Mathematics

Conditional Statements
Many theorems are written in the if-then form “if p, then q,” which is denoted by
p→q Conditional statement

where p is the hypothesis and q is the conclusion. Here are some other ways to express
the conditional statement p → q.
p implies q. p, only if q. p is sufficient for q.
Conditional statements can be either true or false. The conditional statement p → q
is false only when p is true and q is false. To show that a conditional statement is true,
you must prove that the conclusion follows for all cases that fulfill the hypothesis.
To show that a conditional statement is false, you need to describe only a single
counterexample that shows that the statement is not always true.
For instance, x ⫽ ⫺4 is a counterexample that shows that the following statement
is false.
If x2 ⫽ 16, then x ⫽ 4.
The hypothesis “x2 ⫽ 16” is true because 共⫺4兲2 ⫽ 16. However, the conclusion “x ⫽ 4”
is false. This implies that the given conditional statement is false.
For the conditional statement p → q, there are three important associated
conditional statements.
1. The converse of p → q: q → p
2. The inverse of p → q: ~p → ~q
3. The contrapositive of p → q: ~q → ~p
The symbol ~ means the negation of a statement. For instance, the negation of “The
engine is running” is “The engine is not running.”

Example 1 Writing the Converse, Inverse, and Contrapositive

Write the converse, inverse, and contrapositive of the conditional statement “If I get a
B on my test, then I will pass the course.”

Solution
a. Converse: If I pass the course, then I got a B on my test.
b. Inverse: If I do not get a B on my test, then I will not pass the course.
c. Contrapositive: If I do not pass the course, then I did not get a B on my test.

In the example above, notice that neither the converse nor the inverse is logically
equivalent to the original conditional statement. On the other hand, the contrapositive
is logically equivalent to the original conditional statement.

Biconditional Statements
Recall that a conditional statement is a statement of the form “if p, then q.”
A statement of the form “p if and only if q” is called a biconditional statement. A
biconditional statement, denoted by
p↔q Biconditional statement

is the conjunction of the conditional statement p → q and its converse q → p.

A biconditional statement can be either true or false. To be true, both
the conditional statement and its converse must be true.

Example 2 Analyzing a Biconditional Statement

Consider the statement x ⫽ 3 if and only if x2 ⫽ 9.
a. Is the statement a biconditional statement? b. Is the statement true?

Solution
a. The statement is a biconditional statement because it is of the form “p if and only
if q.”
b. The statement can be rewritten as the following conditional statement and its
converse.
Conditional statement: If x ⫽ 3, then x2 ⫽ 9.
Converse: If x2 ⫽ 9, then x ⫽ 3.
The first of these statements is true, but the second is false because x could also
equal ⫺3. So, the biconditional statement is false.

Knowing how to use biconditional statements is an important tool for reasoning in

mathematics.

Example 3 Analyzing a Biconditional Statement

Determine whether the biconditional statement is true or false. If it is false, provide a
counterexample.
A number is divisible by 5 if and only if it ends in 0.

Solution
The biconditional statement can be rewritten as the following conditional statement and
its converse.
Conditional statement: If a number is divisible by 5, then it ends in 0.
Converse: If a number ends in 0, then it is divisible by 5.
The conditional statement is false. A counterexample is the number 15, which is
divisible by 5 but does not end in 0. So, the biconditional statement is false.

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