97909_01_ch01_p009-019.

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SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 19

y
B Increasing and Decreasing Functions
D
The graph shown in Figure 22 rises from A to B, falls from B to C , and rises again from C
y=ƒ to D. The function f is said to be increasing on the interval 关a, b兴, decreasing on 关b, c兴, and
increasing again on 关c, d兴. Notice that if x 1 and x 2 are any two numbers between a and b
C with x 1 ⬍ x 2 , then f 共x 1 兲 ⬍ f 共x 2 兲. We use this as the defining property of an increasing
f(x™)
function.
A f(x¡)

0 a x¡ x™ b c d x A function f is called increasing on an interval I if

FIGURE 22
f 共x 1 兲 ⬍ f 共x 2 兲 whenever x 1 ⬍ x 2 in I

y It is called decreasing on I if
y=≈
f 共x 1 兲 ⬎ f 共x 2 兲 whenever x 1 ⬍ x 2 in I

In the definition of an increasing function it is important to realize that the inequality

0 x f 共x 1 兲 ⬍ f 共x 2 兲 must be satisfied for every pair of numbers x 1 and x 2 in I with x 1 ⬍ x 2.
You can see from Figure 23 that the function f 共x兲 苷 x 2 is decreasing on the interval
FIGURE 23 共⫺⬁, 0兴 and increasing on the interval 关0, ⬁兲.

1.1 Exercises

1. If f 共x兲 苷 x ⫹ s2 ⫺ x and t共u兲 苷 u ⫹ s2 ⫺ u , is it true (c) Estimate the solution of the equation f 共x兲 苷 ⫺1.
that f 苷 t? (d) On what interval is f decreasing?
(e) State the domain and range of f.
2. If
(f) State the domain and range of t.
x2 ⫺ x
f 共x兲 苷 and t共x兲 苷 x
x⫺1 y

is it true that f 苷 t? g
f
3. The graph of a function f is given. 2
(a) State the value of f 共1兲.
(b) Estimate the value of f 共⫺1兲.
0 2 x
(c) For what values of x is f 共x兲 苷 1?
(d) Estimate the value of x such that f 共x兲 苷 0.
(e) State the domain and range of f.
(f) On what interval is f increasing?
5. Figure 1 was recorded by an instrument operated by the Cali-
y
fornia Department of Mines and Geology at the University
Hospital of the University of Southern California in Los Ange-
les. Use it to estimate the range of the vertical ground accelera-
1
tion function at USC during the Northridge earthquake.
0 1 x 6. In this section we discussed examples of ordinary, everyday
functions: Population is a function of time, postage cost is a
function of weight, water temperature is a function of time.
Give three other examples of functions from everyday life that
4. The graphs of f and t are given. are described verbally. What can you say about the domain and
(a) State the values of f 共⫺4兲 and t共3兲. range of each of your functions? If possible, sketch a rough
(b) For what values of x is f 共x兲 苷 t共x兲? graph of each function.

1. Homework Hints available at stewartcalculus.com

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20 CHAPTER 1 FUNCTIONS AND MODELS

7–10 Determine whether the curve is the graph of a function of x. in words what the graph tells you about this race. Who won the
If it is, state the domain and range of the function. race? Did each runner finish the race?

7. y 8. y
y (m)
1 1
A B C
100
0 1 x 0 1 x

9. y 10. y 0 20 t (s)

1 1

0 1 x 0 1 x 15. The graph shows the power consumption for a day in Septem-
ber in San Francisco. (P is measured in megawatts; t is mea-
sured in hours starting at midnight.)
(a) What was the power consumption at 6 AM? At 6 PM?
(b) When was the power consumption the lowest? When was it
11. The graph shown gives the weight of a certain person as a the highest? Do these times seem reasonable?
function of age. Describe in words how this person’s weight
varies over time. What do you think happened when this P
800
person was 30 years old?
600

200 400

weight 150
200
(pounds)
100

50 0 3 6 9 12 15 18 21 t
Pacific Gas & Electric
0 10 20 30 40 50 60 70 age
(years) 16. Sketch a rough graph of the number of hours of daylight as a
function of the time of year.
12. The graph shows the height of the water in a bathtub as a
function of time. Give a verbal description of what you think 17. Sketch a rough graph of the outdoor temperature as a function
happened. of time during a typical spring day.
height 18. Sketch a rough graph of the market value of a new car as a
(inches)
function of time for a period of 20 years. Assume the car is
well maintained.
15
19. Sketch the graph of the amount of a particular brand of coffee
10
sold by a store as a function of the price of the coffee.
5
20. You place a frozen pie in an oven and bake it for an hour. Then
0 5 10 15 time you take it out and let it cool before eating it. Describe how the
(min) temperature of the pie changes as time passes. Then sketch a
rough graph of the temperature of the pie as a function of time.
13. You put some ice cubes in a glass, fill the glass with cold
water, and then let the glass sit on a table. Describe how the 21. A homeowner mows the lawn every Wednesday afternoon.
temperature of the water changes as time passes. Then sketch a Sketch a rough graph of the height of the grass as a function of
rough graph of the temperature of the water as a function of the time over the course of a four-week period.
elapsed time.
22. An airplane takes off from an airport and lands an hour later at
14. Three runners compete in a 100-meter race. The graph depicts another airport, 400 miles away. If t represents the time in min-
the distance run as a function of time for each runner. Describe utes since the plane has left the terminal building, let x共t兲 be

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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.
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SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 21

the horizontal distance traveled and y共t兲 be the altitude of the 1 u⫹1
35. h共x兲 苷 36. f 共u兲 苷
plane. 4
s x 2 ⫺ 5x 1
(a) Sketch a possible graph of x共t兲. 1⫹
u⫹1
(b) Sketch a possible graph of y共t兲.
(c) Sketch a possible graph of the ground speed. 37. F共 p兲 苷 s2 ⫺ s p
(d) Sketch a possible graph of the vertical velocity.

23. The number N (in millions) of US cellular phone subscribers is 38. Find the domain and range and sketch the graph of the
shown in the table. (Midyear estimates are given.) function h共x兲 苷 s4 ⫺ x 2 .

39–50 Find the domain and sketch the graph of the function.
t 1996 1998 2000 2002 2004 2006
39. f 共x兲 苷 2 ⫺ 0.4x 40. F 共x兲 苷 x 2 ⫺ 2x ⫹ 1
N 44 69 109 141 182 233
4 ⫺ t2
(a) Use the data to sketch a rough graph of N as a function of t. 41. f 共t兲 苷 2t ⫹ t 2 42. H共t兲 苷
2⫺t
(b) Use your graph to estimate the number of cell-phone sub-
scribers at midyear in 2001 and 2005. 43. t共x兲 苷 sx ⫺ 5 44. F共x兲 苷 2x ⫹ 1 ⱍ ⱍ
24. Temperature readings T (in °F) were recorded every two hours
45. G共x兲 苷
3x ⫹ x ⱍ ⱍ 46. t共x兲 苷 x ⫺ x ⱍ ⱍ
from midnight to 2:00 PM in Phoenix on September 10, 2008. x

再
The time t was measured in hours from midnight.
x⫹2 if x ⬍ 0
47. f 共x兲 苷
t 0 2 4 6 8 10 12 14 1⫺x if x 艌 0
T 82 75 74 75 84 90 93 94
48. f 共x兲 苷 再 3 ⫺ 12 x
2x ⫺ 5
if x 艋 2
if x ⬎ 2

再
(a) Use the readings to sketch a rough graph of T as a function
of t. x ⫹ 2 if x 艋 ⫺1
49. f 共x兲 苷
(b) Use your graph to estimate the temperature at 9:00 AM. x2 if x ⬎ ⫺1

再
25. If f 共x兲 苷 3x 2 ⫺ x ⫹ 2, find f 共2兲, f 共⫺2兲, f 共a兲, f 共⫺a兲,
x⫹9 if x ⬍ ⫺3
f 共a ⫹ 1兲, 2 f 共a兲, f 共2a兲, f 共a 2 兲, [ f 共a兲] 2, and f 共a ⫹ h兲.
26. A spherical balloon with radius r inches has volume
50. f 共x兲 苷 ⫺2x ⱍ ⱍ
if x 艋 3
⫺6 if x ⬎ 3
V共r兲 苷 43 ␲ r 3. Find a function that represents the amount of air
required to inflate the balloon from a radius of r inches to a
radius of r ⫹ 1 inches. 51–56 Find an expression for the function whose graph is the
given curve.
27–30 Evaluate the difference quotient for the given function. 51. The line segment joining the points 共1, ⫺3兲 and 共5, 7兲
Simplify your answer.
52. The line segment joining the points 共⫺5, 10兲 and 共7, ⫺10兲
f 共3 ⫹ h兲 ⫺ f 共3兲
27. f 共x兲 苷 4 ⫹ 3x ⫺ x 2,
h 53. The bottom half of the parabola x ⫹ 共 y ⫺ 1兲2 苷 0

f 共a ⫹ h兲 ⫺ f 共a兲 54. The top half of the circle x 2 ⫹ 共 y ⫺ 2兲 2 苷 4

28. f 共x兲 苷 x 3,
h
55. y 56. y
1 f 共x兲 ⫺ f 共a兲
29. f 共x兲 苷 ,
x x⫺a

x⫹3 f 共x兲 ⫺ f 共1兲

30. f 共x兲 苷 , 1 1
x⫹1 x⫺1
0 1 x 0 1 x

31–37 Find the domain of the function.

57–61 Find a formula for the described function and state its
x⫹4 2x 3 ⫺ 5
31. f 共x兲 苷 32. f 共x兲 苷 domain.
x2 ⫺ 9 x ⫹x⫺6
2

57. A rectangle has perimeter 20 m. Express the area of the rect-

33. f 共t兲 苷 s
3
2t ⫺ 1 34. t共t兲 苷 s3 ⫺ t ⫺ s2 ⫹ t angle as a function of the length of one of its sides.

Copyright 2010 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.
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22 CHAPTER 1 FUNCTIONS AND MODELS

58. A rectangle has area 16 m2. Express the perimeter of the rect- 67. In a certain country, income tax is assessed as follows. There is
angle as a function of the length of one of its sides. no tax on income up to $10,000. Any income over $10,000 is
taxed at a rate of 10%, up to an income of $20,000. Any income
59. Express the area of an equilateral triangle as a function of the
over $20,000 is taxed at 15%.
length of a side.
(a) Sketch the graph of the tax rate R as a function of the
60. Express the surface area of a cube as a function of its volume. income I.
(b) How much tax is assessed on an income of $14,000?
61. An open rectangular box with volume 2 m3 has a square base.
On $26,000?
Express the surface area of the box as a function of the length
(c) Sketch the graph of the total assessed tax T as a function of
of a side of the base.
the income I.

62. A Norman window has the shape of a rectangle surmounted by 68. The functions in Example 10 and Exercise 67 are called step
a semicircle. If the perimeter of the window is 30 ft, express functions because their graphs look like stairs. Give two other
the area A of the window as a function of the width x of the examples of step functions that arise in everyday life.
window.
69–70 Graphs of f and t are shown. Decide whether each function
is even, odd, or neither. Explain your reasoning.
69. y 70. y
g
f
f

x x
g

63. A box with an open top is to be constructed from a rectangular

piece of cardboard with dimensions 12 in. by 20 in. by cutting 71. (a) If the point 共5, 3兲 is on the graph of an even function, what
out equal squares of side x at each corner and then folding up other point must also be on the graph?
the sides as in the figure. Express the volume V of the box as a (b) If the point 共5, 3兲 is on the graph of an odd function, what
function of x. other point must also be on the graph?

20 72. A function f has domain 关⫺5, 5兴 and a portion of its graph is

shown.
x x (a) Complete the graph of f if it is known that f is even.
x x (b) Complete the graph of f if it is known that f is odd.
12
x x
x x y

64. A cell phone plan has a basic charge of $35 a month. The plan
includes 400 free minutes and charges 10 cents for each addi-
tional minute of usage. Write the monthly cost C as a function 0 x
_5 5
of the number x of minutes used and graph C as a function of x
for 0 艋 x 艋 600.
65. In a certain state the maximum speed permitted on freeways is
73–78 Determine whether f is even, odd, or neither. If you have a
65 mi兾h and the minimum speed is 40 mi兾h. The fine for vio-
graphing calculator, use it to check your answer visually.
lating these limits is $15 for every mile per hour above the
maximum speed or below the minimum speed. Express the x x2
73. f 共x兲 苷 74. f 共x兲 苷
amount of the fine F as a function of the driving speed x and x ⫹1
2
x ⫹1
4

graph F共x兲 for 0 艋 x 艋 100.

x
66. An electricity company charges its customers a base rate of
$10 a month, plus 6 cents per kilowatt-hour (kWh) for the first
75. f 共x兲 苷
x⫹1
76. f 共x兲 苷 x x ⱍ ⱍ
1200 kWh and 7 cents per kWh for all usage over 1200 kWh.
Express the monthly cost E as a function of the amount x of 77. f 共x兲 苷 1 ⫹ 3x 2 ⫺ x 4 78. f 共x兲 苷 1 ⫹ 3x 3 ⫺ x 5
electricity used. Then graph the function E for 0 艋 x 艋 2000.
Copyright 2010 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.
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SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 23

79. If f and t are both even functions, is f ⫹ t even? If f and t are 80. If f and t are both even functions, is the product ft even? If f
both odd functions, is f ⫹ t odd? What if f is even and t is and t are both odd functions, is ft odd? What if f is even and
odd? Justify your answers. t is odd? Justify your answers.

1.2 Mathematical Models: A Catalog of Essential Functions

A mathematical model is a mathematical description (often by means of a function or an
equation) of a real-world phenomenon such as the size of a population, the demand for a
product, the speed of a falling object, the concentration of a product in a chemical reaction,
the life expectancy of a person at birth, or the cost of emission reductions. The purpose of
the model is to understand the phenomenon and perhaps to make predictions about future
behavior.
Figure 1 illustrates the process of mathematical modeling. Given a real-world problem,
our first task is to formulate a mathematical model by identifying and naming the inde-
pendent and dependent variables and making assumptions that simplify the phenomenon
enough to make it mathematically tractable. We use our knowledge of the physical situation
and our mathematical skills to obtain equations that relate the variables. In situations where
there is no physical law to guide us, we may need to collect data (either from a library or
the Internet or by conducting our own experiments) and examine the data in the form of a
table in order to discern patterns. From this numerical representation of a function we may
wish to obtain a graphical representation by plotting the data. The graph might even sug-
gest a suitable algebraic formula in some cases.

Real-world Formulate Mathematical Solve Mathematical Interpret Real-world

problem model conclusions predictions

Test

FIGURE 1 The modeling process

The second stage is to apply the mathematics that we know (such as the calculus that will
be developed throughout this book) to the mathematical model that we have formulated in
order to derive mathematical conclusions. Then, in the third stage, we take those mathe-
matical conclusions and interpret them as information about the original real-world phe-
nomenon by way of offering explanations or making predictions. The final step is to test our
predictions by checking against new real data. If the predictions don’t compare well with
reality, we need to refine our model or to formulate a new model and start the cycle again.
A mathematical model is never a completely accurate representation of a physical situ-
ation—it is an idealization. A good model simplifies reality enough to permit mathematical
calculations but is accurate enough to provide valuable conclusions. It is important to real-
ize the limitations of the model. In the end, Mother Nature has the final say.
There are many different types of functions that can be used to model relationships
observed in the real world. In what follows, we discuss the behavior and graphs of these
functions and give examples of situations appropriately modeled by such functions.

Linear Models
The coordinate geometry of lines is reviewed When we say that y is a linear function of x, we mean that the graph of the function is a
in Appendix B. line, so we can use the slope-intercept form of the equation of a line to write a formula for
Copyright 2010 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.