0.

1 Functions 11

y y

9
8 100

7
80
6
5 60
4
3 40

2
20
1
x x

Figure 0.1.20 −3 −2 −1 1 2 3 −10 −5 5 10

✔QUICK CHECK EXERCISES 0.1 (See page 15 for answers.)

√
1. Let f(x) = x + 1 + 4. 4. The accompanying table gives a 5-day forecast of high and
(a) The natural domain of f is . low temperatures in degrees Fahrenheit ( ◦ F).
(b) f(3) = (a) Suppose that x and y denote the respective high and
(c) f (t 2 − 1) = low temperature predictions for each of the 5 days. Is
(d) f(x) = 7 if x = y a function of x? If so, give the domain and range of
(e) The range of f is . this function.
2. Line segments in an xy-plane form “letters” as depicted. (b) Suppose that x and y denote the respective low and high
temperature predictions for each of the 5 days. Is y a
function of x? If so, give the domain and range of this
function.

mon tue wed thurs fri

(a) If the y-axis is parallel to the letter I, which of the letters high 75 71 65 70 73
represent the graph of y = f(x) for some function f ? low 52 56 48 50 52
(b) If the y-axis is perpendicular to the letter I, which of
the letters represent the graph of y = f(x) for some Table Ex-4
function f ?
3. The accompanying figure shows the complete graph of
y = f(x). 5. Let l, w, and A denote the length, width, and area of a
(a) The domain of f is . rectangle, respectively, and suppose that the width of the
(b) The range of f is . rectangle is half the length.
(c) f (−3) = (a) If l is expressed as a function of w, then l = .
(d) f 21 = (b) If A is expressed as a function of l, then A = .
(e) The solutions to f(x) = − 23 are x = and (c) If w is expressed as a function of A, then w = .
x= .

2
1
x
−3 −2 −1 1 2 3
−1
−2
Figure Ex-3
12 Chapter 0 / Before Calculus

EXERCISE SET 0.1 Graphing Utility

1. Use the accompanying graph to answer the following ques- 4. In each part, compare the natural domains of f and g.
tions, making reasonable approximations where needed. x2 + x
(a) For what values of x is y = 1? (a) f(x) = ; g(x) = x
x+1
(b) For what values of x is y = 3? √ √
(c) For what values of y is x = 3? x x+ x √
(b) f(x) = ; g(x) = x
(d) For what values of x is y ≤ 0? x+1
(e) What are the maximum and minimum values of y and
for what values of x do they occur?
y
3 F O C U S O N C O N C E P TS

5. The accompanying graph shows the median income in

2
U.S. households (adjusted for inflation) between 1990
and 2005. Use the graph to answer the following ques-
1
tions, making reasonable approximations where needed.
x (a) When was the median income at its maximum value,
0
and what was the median income when that occurred?
−1 (b) When was the median income at its minimum value,
and what was the median income when that occurred?
−2 (c) The median income was declining during the 2-year
period between 2000 and 2002. Was it declining
−3 more rapidly during the first year or the second year
−3 −2 −1 0 1 2 3 Figure Ex-1 of that period? Explain your reasoning.

2. Use the accompanying table to answer the questions posed Median U.S. Household Income in
in Exercise 1. Thousands of Constant 2005 Dollars
Median U.S. Household Income

48
x −2 −1 0 2 3 4 5 6
46
y 5 1 −2 7 −1 1 0 9
Table Ex-2 44

42
3. In each part of the accompanying figure, determine whether
the graph defines y as a function of x.
1990 1995 2000 2005
y y
Source: U.S. Census Bureau, August 2006.
Figure Ex-5
x x

6. Use the median income graph in Exercise 5 to answer the

following questions, making reasonable approximations
where needed.
(a) (b) (a) What was the average yearly growth of median in-
come between 1993 and 1999?
y y
(b) The median income was increasing during the 6-year
period between 1993 and 1999. Was it increasing
x x more rapidly during the first 3 years or the last 3
years of that period? Explain your reasoning.
(c) Consider the statement: “After years of decline, me-
dian income this year was finally higher than that of
last year.” In what years would this statement have
(c) (d) been correct?
Figure Ex-3
 0.1 Functions 13
√
7. Find f(0), f(2), f(−2), f(3), f( 2 ), and f(3t). 14. A cup of hot coffee sits on a table. You pour in some
cool milk and let it sit for an hour. Sketch a rough graph
 1, x > 3


(a) f(x) = 3x 2 − 2 (b) f(x) = x of the temperature of the coffee as a function of time.
2x, x ≤ 3


8. Find g(3), g(−1), g(π), g(−1.1), and g(t 2 − 1). 15–18 As seen in Example 3, the equation x 2 + y 2 = 25 does
√ not define y as a function of x. Each graph in these exercises
x+1 x + 1, x≥1
(a) g(x) = (b) g(x) = is a portion of the circle x 2 + y 2 = 25. In each case, determine
x−1 3, x<1 whether the graph defines y as a function of x, and if so, give a
formula for y in terms of x. ■
9–10 Find the natural domain and determine the range of each y y
15. 16.
function. If you have a graphing utility, use it to confirm that 5 5
your result is consistent with the graph produced by your graph-
ing utility. [Note: Set your graphing utility in radian mode when
graphing trigonometric functions.] ■ x x
1 x −5 5 −5 5
9. (a) f(x) = (b) F(x) =
x−3 |x|
√ √
(c) g(x) = x 2 − 3 (d) G(x) = x 2 − 2x + 5
−5 −5
1 x2 − 4
(e) h(x) = (f ) H (x) = 17. y 18. y
1 − sin x x−2
5 5
√ √
10. (a) f(x) = 3 − √x (b) F (x) = 4 − x 2
(c) g(x) = 3 + x (d) G(x) = x 3 +√2
(e) h(x) = 3 sin x (f ) H (x) = (sin x)−2 x x
−5 5 −5 5

F O C U S O N C O N C E P TS
−5 −5
11. (a) If you had a device that could record the Earth’s pop-
ulation continuously, would you expect the graph of
population versus time to be a continuous (unbro- 19–22 True–False Determine whether the statement is true or
ken) curve? Explain what might cause breaks in the false. Explain your answer. ■
curve. 19. A curve that crosses the x-axis at two different points cannot
(b) Suppose that a hospital patient receives an injection be the graph of a function.
of an antibiotic every 8 hours and that between in-
20. The natural domain of a real-valued function defined by a
jections the concentration C of the antibiotic in the
formula consists of all those real numbers for which the
bloodstream decreases as the antibiotic is absorbed
formula yields a real value.
by the tissues. What might the graph of C versus
the elapsed time t look like? 21. The range of the absolute value function is all positive real
numbers.
12. (a) If you had a device that could record the tempera- √
ture of a room continuously over a 24-hour period, 22. If g(x) = 1/ f(x), then the domain of g consists of all
would you expect the graph of temperature versus those real numbers x for which f(x)  = 0.
time to be a continuous (unbroken) curve? Explain 23. Use the equation y = x 2 − 6x + 8 to answer the following
your reasoning. questions.
(b) If you had a computer that could track the number (a) For what values of x is y = 0?
of boxes of cereal on the shelf of a market contin- (b) For what values of x is y = −10?
uously over a 1-week period, would you expect the (c) For what values of x is y ≥ 0?
graph of the number of boxes on the shelf versus (d) Does y have a minimum value? A maximum value? If
time to be a continuous (unbroken) curve? Explain so, find them.
your reasoning. √
24. Use the equation y = 1 + x to answer the following ques-
13. A boat is bobbing up and down on some gentle waves. tions.
Suddenly it gets hit by a large wave and sinks. Sketch (a) For what values of x is y = 4?
a rough graph of the height of the boat above the ocean (b) For what values of x is y = 0?
floor as a function of time. (c) For what values of x is y ≥ 6? (cont.)
14 Chapter 0 / Before Calculus

(d) Does y have a minimum value? A maximum value? If (d) Plot the function in part (b) and estimate the dimensions
so, find them. of the enclosure that minimize the amount of fencing
25. As shown in the accompanying figure, a pendulum of con- required.
stant length L makes an angle θ with its vertical position. 32. As shown in the accompanying figure, a camera is mounted
Express the height h as a function of the angle θ. at a point 3000 ft from the base of a rocket launching pad.
26. Express the length L of a chord of a circle with radius 10 cm The rocket rises vertically when launched, and the camera’s
as a function of the central angle θ (see the accompanying elevation angle is continually adjusted to follow the bottom
figure). of the rocket.
(a) Express the height x as a function of the elevation an-
gle θ.
L
(b) Find the domain of the function in part (a).
u
u 10 cm (c) Plot the graph of the function in part (a) and use it to
L estimate the height of the rocket when the elevation an-
gle is π/4 ≈ 0.7854 radian. Compare this estimate to
h
the exact height.
Figure Ex-25 Figure Ex-26
Rocket
27–28 Express the function in piecewise form without using
absolute values. [Suggestion: It may help to generate the graph
of the function.] ■
x
27. (a) f(x) = |x| + 3x + 1 (b) g(x) = |x| + |x − 1|
28. (a) f(x) = 3 + |2x − 5| (b) g(x) = 3|x − 2| − |x + 1|
u
29. As shown in the accompanying figure, an open box is to
3000 ft
be constructed from a rectangular sheet of metal, 8 in by 15 Camera Figure Ex-32
in, by cutting out squares with sides of length x from each
corner and bending up the sides. 33. A soup company wants to manufacture a can in the shape
(a) Express the volume V as a function of x. of a right circular cylinder that will hold 500 cm3 of liquid.
(b) Find the domain of V . The material for the top and bottom costs 0.02 cent/cm2 ,
(c) Plot the graph of the function V obtained in part (a) and and the material for the sides costs 0.01 cent/cm2 .
estimate the range of this function. (a) Estimate the radius r and the height h of the can that
(d) In words, describe how the volume V varies with x, and costs the least to manufacture. [Suggestion: Express
discuss how one might construct boxes of maximum the cost C in terms of r.]
volume. (b) Suppose that the tops and bottoms of radius r are
x x punched out from square sheets with sides of length
x x
2r and the scraps are waste. If you allow for the cost of
8 in the waste, would you expect the can of least cost to be
x x taller or shorter than the one in part (a)? Explain.
x x
15 in (c) Estimate the radius, height, and cost of the can in part
Figure Ex-29
(b), and determine whether your conjecture was correct.
34. The designer of a sports facility wants to put a quarter-mile
30. Repeat Exercise 29 assuming the box is constructed in the (1320 ft) running track around a football field, oriented as
same fashion from a 6-inch-square sheet of metal. in the accompanying figure on the next page. The football
31. A construction company has adjoined a 1000 ft2 rectan- field is 360 ft long (including the end zones) and 160 ft wide.
gular enclosure to its office building. Three sides of the The track consists of two straightaways and two semicircles,
enclosure are fenced in. The side of the building adjacent with the straightaways extending at least the length of the
to the enclosure is 100 ft long and a portion of this side is football field.
used as the fourth side of the enclosure. Let x and y be the (a) Show that it is possible to construct a quarter-mile track
dimensions of the enclosure, where x is measured parallel around the football field. [Suggestion: Find the shortest
to the building, and let L be the length of fencing required track that can be constructed around the field.]
for those dimensions. (b) Let L be the length of a straightaway (in feet), and let x
(a) Find a formula for L in terms of x and y. be the distance (in feet) between a sideline of the foot-
(b) Find a formula that expresses L as a function of x alone. ball field and a straightaway. Make a graph of L ver-
(c) What is the domain of the function in part (b)? sus x. (cont.)
 0.2 New Functions from Old 15

(c) Use the graph to estimate the value of x that produces ature T and wind speed v, the wind chill temperature index
the shortest straightaways, and then find this value of x is the equivalent temperature that exposed skin would feel
exactly. with a wind speed of v mi/h. Based on a more accurate
(d) Use the graph to estimate the length of the longest pos- model of cooling due to wind, the new formula is
sible straightaways, and then find that length exactly. T, 0≤v≤3
WCT =
35.74 + 0.6215T − 35.75v 0.16 + 0.4275T v 0.16 , 3<v
◦
where T is the temperature in F, v is the wind speed in
mi/h, and WCT is the equivalent temperature in ◦ F. Find
160′ the WCT to the nearest degree if T = 25 ◦ F and
(a) v = 3 mi/h (b) v = 15 mi/h (c) v = 46 mi/h.
Source: Adapted from UMAP Module 658, Windchill, W. Bosch and
L. Cobb, COMAP, Arlington, MA.
360′
Figure Ex-34
38–40 Use the formula for the wind chill temperature index
described in Exercise 37. ■
35–36 (i) Explain why the function f has one or more holes 38. Find the air temperature to the nearest degree if the WCT is
in its graph, and state the x-values at which those holes occur. reported as −60 ◦ F with a wind speed of 48 mi/h.
(ii) Find a function g whose graph is identical to that of f, but
39. Find the air temperature to the nearest degree if the WCT is
without the holes. ■
reported as −10 ◦ F with a wind speed of 48 mi/h.
(x + 2)(x 2 − 1) x 2 + |x|
35. f(x) = 36. f(x) = 40. Find the wind speed to the nearest mile per hour if the WCT
(x + 2)(x − 1) |x| is reported as 5 ◦ F with an air temperature of 20 ◦ F.
37. In 2001 the National Weather Service introduced a new wind
chill temperature (WCT) index. For a given outside temper-

✔QUICK CHECK ANSWERS 0.1

1. (a) [−1, +⬁) (b) 6 (c) |t| + 4 (d) 8 (e) [4, +⬁) 2. (a) M (b) I 3. (a) [−3, 3) (b) [−2, 2] (c) −1 (d) 1
(e) − 43 ; − 23 4. (a) yes; domain: {65, 70, 71, 73, 75}; range: {48, 50, 52, 56} (b) no 5. (a) l = 2w (b) A = l 2 /2
√
(c) w = A/2

0.2 NEW FUNCTIONS FROM OLD

Just as numbers can be added, subtracted, multiplied, and divided to produce other
numbers, so functions can be added, subtracted, multiplied, and divided to produce other
functions. In this section we will discuss these operations and some others that have no
analogs in ordinary arithmetic.

ARITHMETIC OPERATIONS ON FUNCTIONS

Two functions, f and g, can be added, subtracted, multiplied, and divided in a natural way
to form new functions f + g, f − g, f g, and f /g. For example, f + g is defined by the
formula
(f + g)(x) = f(x) + g(x) (1)
which states that for each input the value of f + g is obtained by adding the values of
f and g. Equation (1) provides a formula for f + g but does not say anything about the
domain of f + g. However, for the right side of this equation to be defined, x must lie in
the domains of both f and g, so we define the domain of f + g to be the intersection of
these two domains. More generally, we make the following definition.