Poor index with dividends reinvested

3

159

Functions

1,500 P

y

1,000

500

1970 1975 1980 1985 1990 1995 2000 2005 2010 0

Figure 1 Standard and Poor’s Index with dividends reinvested (credit “bull”: modification of work by Prayitno hadinata; credit “graph”: modification of work by measuringWorth)

Introduction Toward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. Figure 1 tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.

Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.

ChAPTeR OUTlIne

3.1 Functions and Function notation 3.2 domain and Range 3.3 Rates of Change and behavior of graphs 3.4 Composition of Functions 3.5 Transformation of Functions 3.6 Absolute value Functions 3.7 Inverse Functions

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

176 CHAPTER 3 fuNctioNs

3.1 SeCTIOn exeRCISeS

veRbAl

1. What is the difference between a relation and a function?

2. What is the difference between the input and the output of a function?

3. Why does the vertical line test tell us whether the graph of a relation represents a function?

4. How can you determine if a relation is a one-to-one function?

5. Why does the horizontal line test tell us whether the graph of a function is one-to-one?

AlgebRAIC

For the following exercises, determine whether the relation represents a function.

6. {(a, b), (c, d), (a, c)} 7. {(a, b),(b, c),(c, c)}

For the following exercises, determine whether the relation represents y as a function of x. 8. 5x + 2y = 10 9. y = x 2 10. x = y 2

11. 3x 2 + y = 14 12. 2x + y 2 = 6 13. y = −2x 2 + 40x

14. y = 1 __ x 15. x = 3y + 5

_ 7y − 1 16. x = √

— 1 − y 2

17. y = 3x + 5 ______ 7x − 1 18. x 2 + y 2 = 9 19. 2xy = 1

20. x = y 3 21. y = x 3 22. y = √ —

1 − x 2

23. x = ± √ —

1 − y 24. y = ± √ —

1 − x 25. y 2 = x 2

26. y 3 = x 2

For the following exercises, evaluate the function f at the indicated values f (−3), f (2), f (−a), −f (a), f (a + h).

27. f (x) = 2x − 5 28. f (x) = −5x 2 + 2x − 1 29. f (x) = √ —

2 − x + 5

30. f (x) = 6x − 1 ______ 5x + 2 31. f (x) = ∣ x − 1 ∣ − ∣ x + 1 ∣

32. Given the function g(x) = 5 − x 2, simplify g(x + h) − g(x)

__ h

, h ≠ 0

33. Given the function g(x) = x 2 + 2x, simplify g(x) − g(a)

_ x − a , x ≠ a

34. Given the function k(t) = 2t − 1: a. Evaluate k(2). b. Solve k(t) = 7.

35. Given the function f (x) = 8 − 3x: a. Evaluate f (−2). b. Solve f (x) = −1.

36. Given the function p(c) = c 2 + c: a. Evaluate p(−3). b. Solve p(c) = 2.

37. Given the function f (x) = x 2 − 3x a. Evaluate f (5). b. Solve f (x) = 4

38. Given the function f (x) =  √ —

x + 2 : a. Evaluate f (7). b. Solve f (x) = 4

39. Consider the relationship 3r + 2t = 18. a. Write the relationship as a function r = f (t). b. Evaluate f (−3). c. Solve f (t) = 2.

 

 

SECTION 3.1 sectioN exercises 177

gRAPhICAl

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

40.

x

y 41.

x

y 42.

x

y

43.

x

y 44. 45.

x

y

46.

x

y 47.

x

y 48.

x

y

49.

x

y 50.

x

y 51.

x

y

x

y

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

178 CHAPTER 3 fuNctioNs

52. Given the following graph a. Evaluate f (−1). b. Solve for f (x) = 3.

53. Given the following graph a. Evaluate f (0). b. Solve for f (x) = −3.

54. Given the following graph a. Evaluate f (4). b. Solve for f (x) = 1.

For the following exercises, determine if the given graph is a one-to-one function. 55.

x

y 56.

x

y 57.

x

y

58.

x

y 59.

x

y

π�π

5 4 3 2 1

�1 �2 �3 �4 �5

nUmeRIC For the following exercises, determine whether the relation represents a function.

60. {(−1, −1),(−2, −2),(−3, −3)} 61. {(3, 4),(4, 5),(5, 6)} 62. {(2, 5),(7, 11),(15, 8),(7, 9)}

For the following exercises, determine if the relation represented in table form represents y as a function of x.

63. x 5 10 15 y 3 8 14

64. x 5 10 15 y 3 8 8

65. x 5 10 10 y 3 8 14

For the following exercises, use the function f represented in Table 14 below.

x 0 1 2 3 4 5 6 7 8 9 f (x) 74 28 1 53 56 3 36 45 14 47

Table 14

66. Evaluate f (3). 67. Solve f (x) = 1

x

y

x

y

x

y

 

 

SECTION 3.1 sectioN exercises 179

For the following exercises, evaluate the function f at the values f (−2), f (−1), f (0), f (1), and f (2).

68. f (x) = 4 − 2x 69. f (x) = 8 − 3x 70. f (x) = 8x 2 − 7x + 3

71. f (x) = 3 + √ —

x + 3 72. f (x) = x − 2 _ x + 3 73. f (x) = 3x

For the following exercises, evaluate the expressions, given functions f, g, and h:

f (x) = 3x − 2 g(x) = 5 − x2 h(x) = −2×2 + 3x − 1

74. 3f (1) − 4g(−2) 75. f  7 _ 3  − h(−2)

TeChnOlOgy

For the following exercises, graph y = x2 on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

76. [−0.1, 0.1] 77. [−10, 10] 78. [−100, 100]

For the following exercises, graph y = x3 on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

79. [−0.1, 0.1] 80. [−10, 10] 81. [−100, 100]

For the following exercises, graph y = √ — x on the given viewing window. Determine the corresponding range for each

viewing window. Show each graph. 82. [0, 0.01] 83. [0, 100] 84. [0, 10,000]

For the following exercises, graph y = 3 √ — x on the given viewing window. Determine the corresponding range for each

viewing window. Show each graph. 85. [−0.001, 0.001] 86. [−1,000, 1,000] 87. [−1,000,000, 1,000,000]

ReAl-WORld APPlICATIOnS 88. The amount of garbage, G, produced by a city with

population p is given by G = f (p). G is measured in tons per week, and p is measured in thousands of people. a. The town of Tola has a population of 40,000 and

produces 13 tons of garbage each week. Express this information in terms of the function f.

b. Explain the meaning of the statement f (5) = 2.

89. The number of cubic yards of dirt, D, needed to cover a garden with area a square feet is given by D = g(a). a. A garden with area 5,000 ft2 requires 50 yd3 of dirt.

Express this information in terms of the function g. b. Explain the meaning of the statement g(100) = 1.

90. Let f (t) be the number of ducks in a lake t years after 1990. Explain the meaning of each statement: a. f (5) = 30 b. f (10) = 40

91. Let h(t) be the height above ground, in feet, of a rocket t seconds after launching. Explain the meaning of each statement: a. h(1) = 200 b. h(2) = 350

92. Show that the function f (x) = 3(x − 5)2 + 7 is not one-to-one.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

SECTION 3.2 sectioN exercises 193

3.2 SeCTIOn exeRCISeS

veRbAl

1. Why does the domain differ for different functions? 2. How do we determine the domain of a function defined by an equation?

3. Explain why the domain of f (x) = 3 √ — x is different

from the domain of f (x) = √ — x .

4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

5. How do you graph a piecewise function?

AlgebRAIC

For the following exercises, find the domain of each function using interval notation.

6. f (x) = −2x(x − 1)(x − 2) 7. f (x) = 5 − 2×2 8. f (x) = 3 √ —

x − 2

9. f (x) = 3 − √ —

6 − 2x 10. f (x) = √ —

4 − 3x 11. f (x) = √ —

x 2 + 4

12. f (x) =   3 √ —

1 − 2x 13. f (x) =   3 √ —

x − 1 14. f (x) =   9 _____ x − 6

15. f (x) =  3x + 1 ______ 4x + 2 16. f (x) =   √ —

x + 4 _______ x − 4 17. f (x) =   x − 3 __________ x2 + 9x − 22

18. f (x) =   1 ________ x2 − x − 6 19. f (x) =   2×3 − 250 __________ x2 − 2x − 15

20. f (x) =   5 _ √ —

x − 3

21. f (x) = 2x + 1 _ √ —

5 − x 22. f (x) = √

— x − 4 _

√ —

x − 6 23. f (x) = √

— x − 6 _

√ —

x − 4

24. f (x) =  x __ x 25. f (x) = x 2 − 9x _

x2 − 81

26. Find the domain of the function f (x) = √ —

2x 3 − 50x by: a. using algebra. b. graphing the function in the radicand and determining intervals on the x-axis for which the radicand is

nonnegative.

gRAPhICAl

For the following exercises, write the domain and range of each function using interval notation.

27.

4

x

y

–2–2–4

–4

–6

–6

–8

–8

–10

–10

2

6

6

42 8

8

10

10

28.

4

x

y

–2–2–4

–4

–6

–6

–8

–8

–10

–10

2

6

6

42 8

8

10

10

29.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

194 CHAPTER 3 fuNctioNs

30.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

31.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

32.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

33.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

34.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

35.

1 6

–6, – )(

1 6

6, )(

1 6

, )( –6–

1 6

, )( 6

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5–6

–5 –6

1

3

3

21 4

4

5 6

5 6

36.

4

x

y

–2–2–4

–4

–6

–6

–8

–8

–10

–10

2

6

6

42 8

8

10

10

37.

4

x

y

–2–2–4

–4

–6

–6

–8

–8

–10

–10

2

6

6

42 8

8

10

10

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

38. f (x) =  x + 1 if x < −2

−2x − 3 if x ≥ −2 { 39. f (x) =  2x − 1 if x < 1 1 + x if x ≥ 1 { 40. f (x) =  x + 1 if x < 0 x − 1 if x > 0 {

41. f (x) =  3 if x < 0 √

— x if x ≥ 0 { 42. f (x) =  x

2 if x < 0 1 − x if x > 0 { 43. f (x) =  x

2 if x < 0 x + 2 if x ≥ 0 {

44. f (x) =  x + 1 if x < 1 x3 if x ≥ 1 { 45. f (x) =  | x | if x < 2 1 if x ≥ 2 {

 

 

SECTION 3.2 sectioN exercises 195

nUmeRIC For the following exercises, given each function f, evaluate f (−3), f (−2), f (−1), and f (0).

46. f (x) =  x + 1 if x < −2

−2x − 3 if x ≥ −2 { 47. f (x) =  1 if x ≤ −3 0 if x > −3 { 48. f (x) =  −2x 2 + 3 if x ≤ −1

5x − 7 if x > −1 {

For the following exercises, given each function f, evaluate f (−1), f (0), f (2), and f (4).

49. f (x) =  7x + 3 if x < 0 7x + 6 if x ≥ 0{ 50. f (x) =  x

2 − 2 if x < 2 4 + | x − 5 | if x ≥ 2{ 51. f (x) =

5x if x < 0 3 if 0 ≤ x ≤ 3 x 2 if x > 3{

For the following exercises, write the domain for the piecewise function in interval notation.

52. f (x) =  x + 1 if x < −2

−2x − 3 if x ≥ −2 { 53. f (x) =  x 2 − 2 if x < 1

−x2 + 2 if x > 1 { 54. f (x) =  2x − 3 if x < 0 −3×2 if x ≥ 2 {

TeChnOlOgy

55. Graph y = 1 __ x 2 on the viewing window [−0.5, −0.1] and [0.1, 0.5]. Determine the corresponding range for the viewing window. Show the graphs.

56. Graph y = 1 _ x on the viewing window [−0.5, −0.1] and [0.1, 0.5]. Determine the corresponding range for the viewing window. Show the graphs.

exTenSIOn 57. Suppose the range of a function f is [−5, 8]. What is the range of | f (x) |?

58. Create a function in which the range is all nonnegative real numbers.

59. Create a function in which the domain is x > 2.

ReAl-WORld APPlICATIOnS

60. The height h of a projectile is a function of the time t it is in the air. The height in feet for t seconds is given by the function h(t) = −16t 2 + 96t. What is the domain of the function? What does the domain mean in the context of the problem?

61. The cost in dollars of making x items is given by the function C(x) = 10x + 500. a. The fixed cost is determined when zero items are

produced. Find the fixed cost for this item. b. What is the cost of making 25 items? c. Suppose the maximum cost allowed is $1500. What

are the domain and range of the cost function, C(x)?

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

CHAPTER 3 fuNctioNs206

3.3 SeCTIOn exeRCISeS

veRbAl

1. Can the average rate of change of a function be constant?

2. If a function f is increasing on (a, b) and decreasing on (b, c), then what can be said about the local extremum of f on (a, c)?

3. How are the absolute maximum and minimum similar to and different from the local extrema?

4. How does the graph of the absolute value function compare to the graph of the quadratic function, y = x2, in terms of increasing and decreasing intervals?

AlgebRAIC For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

5. f (x) = 4x 2 − 7 on [1, b] 6. g (x) = 2x 2 − 9 on [4, b]

7. p(x) = 3x + 4 on [2, 2 + h] 8. k(x) = 4x − 2 on [3, 3 + h]

9. f (x) = 2x 2 + 1 on [x, x + h] 10. g(x) = 3×2 − 2 on [x, x + h]

11. a(t) = 1 ____ t + 4 on [9, 9 + h] 12. b(x) = 1 _____ x + 3 on [1, 1 + h]

13. j(x) = 3x 3 on [1, 1 + h] 14. r(t) = 4t 3 on [2, 2 + h]

15. f (x + h) − f(x)

__ h

given f(x) = 2×2 − 3x on [x, x + h]

gRAPhICAl

For the following exercises, consider the graph of f shown in Figure 15.

16. Estimate the average rate of change from x = 1 to x = 4.

17. Estimate the average rate of change from x = 2 to x = 5.

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

18.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5 19.

0

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

20.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

Figure 15

x

y

1

5

6

7

3

2

– 7 8321 654

4

 

 

SECTION 3.3 sectioN exercises 207

21.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5–6

–5 –6

1

3

3

21 4

4

5 6

5 6

For the following exercises, consider the graph shown in Figure 16.

22. Estimate the intervals where the function is increasing or decreasing.

23. Estimate the point(s) at which the graph of f has a local maximum or a local minimum.

40

x

y

–1 –20–2

–40

–3

–60

–4

–80

–5

–100

20

3

60

21 4

80

5

100

Figure 16

For the following exercises, consider the graph in Figure 17.

24. If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.

25. If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

100

x

y

–2 –50–4

–100

–6

–150

–8

–200

–10

–250

50

6

150

42 8

200

10

250

Figure 17

nUmeRIC

26. Table 3 gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?

Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 Sales (millions of dollars) 201 219 233 243 249 251 249 243 233

Table 3

27. Table 4 gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 Population (thousands) 87 84 83 80 77 76 78 81 85

Table 4

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

208 CHAPTER 3 fuNctioNs

For the following exercises, find the average rate of change of each function on the interval specified. 28. f (x) = x 2 on [1, 5] 29. h(x) = 5 − 2x 2 on [−2, 4]

30. q(x) = x3 on [−4, 2] 31. g (x) = 3x 3 − 1 on [−3, 3]

32. y = 1 _ x on [1, 3] 33. p(t) = (t 2 − 4)(t + 1) ____________ t 2 + 3 on [−3, 1]

34. k (t) = 6t2 + 4 __ t3 on [−1, 3]

TeChnOlOgy For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

35. f (x) = x 4 − 4x 3 + 5 36. h(x) = x 5 + 5x 4 + 10x 3 + 10x 2 − 1

37. g(t) = t √ —

t + 3 38. k(t) = 3 t 2 _ 3 − t

39. m(x) = x 4 + 2x 3 − 12x 2 − 10x + 4 40. n(x) = x 4 − 8x 3 + 18x 2 − 6x + 2

exTenSIOn 41. The graph of the function f is shown in Figure 18.

Maximum X = 1.3333324 Y = 5.1851852

 

Figure 18

Based on the calculator screen shot, the point (1.333, 5.185) is which of the following?

a. a relative (local) maximum of the function b. the vertex of the function c. the absolute maximum of the function d. a zero of the function

42. Let f (x) = 1 __ x . Find a number c such that the average rate of change of the function f on the interval (1, c) is −  1 __ 4

43. Let f(x) =  1 __ x . Find the number b such that the average rate of change of f on the interval (2, b) is − 1 __ 10 .

ReAl-WORld APPlICATIOnS 44. At the start of a trip, the odometer on a car read

21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?

45. A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

46. Near the surface of the moon, the distance that an object falls is a function of time. It is given by d(t) = 2.6667t2, where t is in seconds and d(t) is in feet. If an object is dropped from a certain height, find the average velocity of the object from t = 1 to t = 2.

47. The graph in Figure 19 illustrates the decay of a radioactive substance over t days.

0

8

t

A

6

10

10

5 15 Time (days)A

m ou

nt (m

ill ig

ra m

s)

14 12

20

16

Figure 19

Use the graph to estimate the average decay rate from t = 5 to t = 15.

 

 

218 CHAPTER 3 fuNctioNs

3.4 SeCTIOn exeRCISeS

veRbAl

1. How does one find the domain of the quotient of

two functions, f _ g ?

2. What is the composition of two functions, f ∘ g ?

3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition ? If yes, give an example. If no, explain why not.

4. How do you find the domain for the composition of two functions, f ∘ g ?

AlgebRAIC

For the following exercises, determine the domain for each function in interval notation.

5. Given f (x) = x 2 + 2x and g(x) = 6 − x 2, find f + g,

f − g, fg, and f _ g .

6. Given f (x) = −3×2 + x and g(x) = 5, find f + g,

f − g, fg, and f _ g .

7. Given f (x) = 2x 2 + 4x and g(x) = 1 _ 2x , find f + g,

f − g, fg, and f _ g .

8. Given f (x) = 1 _ x − 4 and g(x) = 1 _

6 − x , find

f + g, f − g, fg, and f _ g .

9. Given f (x) = 3x 2 and g(x) = √ —

x − 5 , find f + g,

f − g, fg, and f _ g .

10. Given f (x) = √ — x and g(x) = |x − 3|, find

g _

f .

11. For the following exercise, find the indicated function given f (x) = 2x 2 + 1 and g(x) = 3x − 5. a. f ( g(2)) b. f ( g(x)) c. g( f (x)) d. ( g ∘ g)(x) e. ( f ∘ f )(−2)

For the following exercises, use each pair of functions to find f (g(x)) and g(f (x)). Simplify your answers.

12. f (x) = x 2 + 1, g(x) = √ —

x + 2 13. f (x) = √ — x + 2, g(x) = x 2 + 3

14. f (x) = |x|, g(x) = 5x + 1 15. f (x) = 3 √— x , g(x) = x + 1 _ x3

 

16. f (x) = 1 _ x − 6

, g(x) = 7 _ x + 6 17. f (x) = 1 ___ x−4 , g(x) =

2 _ x + 4

For the following exercises, use each set of functions to find f (g(h(x))). Simplify your answers.

18. f (x) = x 4 + 6, g(x) = x − 6, and h(x) = √ — x 19. f (x) = x 2 + 1, g(x) =

1 _ x , and h(x) = x + 3

20. Given f (x) = 1 _ x , and g(x) = x − 3, find the following: a. ( f ∘ g)(x) b. the domain of ( f ∘ g)(x) in interval notation c. ( g ∘ f )(x) d. the domain of ( g ∘ f )(x)

e.  f _ g  x

21. Given f (x) = √ —

2 − 4x and g(x) = −  3 _ x , find the following: a. ( g ∘ f )(x) b. the domain of ( g ∘ f )(x) in interval notation

 

 

SECTION 3.4 sectioN exercises 219

22. Given the functions f (x) = 1 − x _ x and g(x) = 1 _

1 + x2 ,

find the following:

a. ( g ∘ f )(x) b. ( g ∘ f )(2)

23. Given functions p(x) =   1 _ √

— x and m(x) = x 2 − 4,

state the domain of each of the following functions using interval notation:

a. p(x)

_ m(x)

 

b. p(m(x)) c. m(p(x))

24. Given functions q(x) = 1 _ √

— x and h(x) = x 2 − 9, state

the domain of each of the following functions using interval notation.

a. q(x)

_ h(x)

 

b. q(h(x)) c. h(q(x))

25. For f (x) = 1 _ x and g(x) = √ —

x − 1 , write the domain

of ( f ∘ g)(x) in interval notation.

For the following exercises, find functions f (x) and g(x) so the given function can be expressed as h(x) = f (g(x)).

26. h(x) = (x + 2)2 27. h(x) = (x − 5)3 28. h(x) = 3 _ x − 5 29. h(x) = 4 _______ (x + 2)2

30. h(x) = 4 + 3 √ — x 31. h(x) =

3 √

_______

1 ______ 2x − 3 32. h(x) = 1 _

(3x 2 − 4)−3 33. h(x) =

4 √

_______

3x − 2 ______ x + 5

34. h(x) =  8 + x 3 _ 8 − x 3  4

35. h(x) = √ —

2x + 6 36. h(x) = (5x − 1)3 37. h(x) = 3 √ —

x − 1

38. h(x) = |x 2 + 7| 39. h(x) = 1 _ (x − 2)3

40. h(x) =  1 _ 2x − 3  2

41. h(x) = √ _______

2x − 1 _ 3x + 4

gRAPhICAl

For the following exercises, use the graphs of f, shown in Figure 4, and g, shown in Figure 5, to evaluate the expressions.

f(x)

x –1

1

1–

5

6

3

2

– 7321 654

4 f

Figure 4

f(x)

x –1

1

1–

5

6

3

2

– 7321 654

4 g

Figure 5

42. f ( g(3)) 43. f ( g(1)) 44. g( f (1)) 45. g( f (0))

46. f ( f (5)) 47. f ( f (4)) 48. g( g(2)) 49. g( g(0))

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

220 CHAPTER 3 fuNctioNs

For the following exercises, use graphs of f (x), shown in Figure 6, g(x), shown in Figure 7, and h(x), shown in Figure 8, to evaluate the expressions.

f(x)

x

f (x)

– –1

1

1

–

2

–

3

–

4

3 24

5

3 421

Figure 6

g(x)

x

f (x)

– –1

1

1

–

2

–

3

–

4

3 24

5

3 421

Figure 7

h(x)

x

f (x)

– –1

1

1

–

2

–

3

–

4

3 24

5

3 421

Figure 8

50. g( f (1)) 51. g( f (2)) 52. f ( g(4)) 53. f ( g(1))

54. f (h(2)) 55. h( f (2)) 56. f ( g(h(4))) 57. f ( g( f (−2)))

nUmeRIC

For the following exercises, use the function values for f and g shown in Table 3 to evaluate each expression.

x 0 1 2 3 4 5 6 7 8 9 f (x) 7 6 5 8 4 0 2 1 9 3 g(x) 9 5 6 2 1 8 7 3 4 0

Table 3

58. f ( g(8)) 59. f ( g(5)) 60. g( f (5)) 61. g( f (3)) 62. f ( f (4)) 63. f ( f (1)) 64. g( g(2)) 65. g( g(6))

For the following exercises, use the function values for f and g shown in Table 4 to evaluate the expressions.

x −3 −2 −1 0 1 2 3 f (x) 11 9 7 5 3 1 −1 g(x) −8 −3 0 1 0 −3 −8

Table 4

66. ( f ∘ g)(1) 67. ( f ∘ g)(2) 68. ( g ∘ f )(2) 69. ( g ∘ f )(3) 70. ( g ∘ g )(1) 71. ( f ∘ f )(3)

For the following exercises, use each pair of functions to find f (g(0)) and g(f (0)).

72. f (x) = 4x + 8, g(x) = 7 − x2 73. f (x) = 5x + 7, g(x) = 4 − 2×2

74. f (x) = √ —

x + 4 , g(x) = 12 − x3 75. f (x) = 1 _ x + 2 , g(x) = 4x + 3

For the following exercises, use the functions f (x) = 2×2 + 1 and g(x) = 3x + 5 to evaluate or find the composite function as indicated.

76. f ( g(2)) 77. f ( g(x)) 78. g( f ( − 3)) 79. ( g ∘ g )(x)

 

 

SECTION 3.4 sectioN exercises 221

exTenSIOnS

For the following exercises, use f (x) = x3 + 1 and g(x) = 3 √— x − 1 .

80. Find ( f ∘ g)(x) and ( g ∘ f )(x). Compare the two answers. 81. Find ( f ∘ g)(2) and ( g ∘ f )(2).

82. What is the domain of ( g ∘ f )(x) ? 83. What is the domain of ( f ∘ g)(x) ?

84. Let f (x) = 1 __ x .

a. Find ( f ∘ f )(x). b. Is ( f ∘ f )(x) for any function f the same result as the answer to part (a) for any function ? Explain.

For the following exercises, let F (x) = (x + 1)5, f (x) = x5, and g(x) = x + 1.

85. True or False: ( g ∘ f )(x) = F (x). 86. True or False: ( f ∘ g )(x) = F (x).

For the following exercises, find the composition when f (x) = x2 + 2 for all x ≥ 0 and g(x) = √ —

x − 2 .

87. ( f ∘ g)(6) ; ( g ∘ f )(6) 88. ( g ∘ f )(a) ; ( f ∘ g )(a) 89. ( f ∘ g )(11) ; (g ∘ f )(11)

ReAl-WORld APPlICATIOnS 90. The function D(p) gives the number of items

that will be demanded when the price is p. The production cost C(x) is the cost of producing x items. To determine the cost of production when the price is $6, you would do which of the following ? a. Evaluate D(C(6)). b. Evaluate C(D(6)). c. Solve D(C(x)) = 6. d. Solve C(D(p)) = 6.

91. The function A(d) gives the pain level on a scale of 0 to 10 experienced by a patient with d milligrams of a pain- reducing drug in her system. The milligrams of the drug in the patient’s system after t minutes is modeled by m(t). Which of the following would you do in order to determine when the patient will be at a pain level of 4 ? a. Evaluate A(m(4)). b. Evaluate m(A(4)). c. Solve A(m(t)) = 4. d. Solve m(A(d)) = 4.

92. A store offers customers a 30 % discount on the price x of selected items. Then, the store takes off an additional 15 % at the cash register. Write a price function P(x) that computes the final price of the item in terms of the original price x. (Hint: Use function composition to find your answer.)

93. A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to r(t) = 25 √— t + 2 , find the area of the ripple as a function of time. Find the area of the ripple at t = 2.

94. A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula r(t) = 2t + 1, express the area burned as a function of time, t (minutes).

95. Use the function you found in the previous exercise to find the total area burned after 5 minutes.

96. The radius r, in inches, of a spherical balloon is related to the volume, V, by r(V) =

3 √

___

3V ___ 4π . Air is pumped into the balloon, so the volume after t seconds is given by V(t) = 10 + 20t. a. Find the composite function r(V(t)). b. Find the exact time when the radius reaches

10 inches.

97. The number of bacteria in a refrigerated food product is given by

N(T) = 23T 2 − 56T + 1, 3 < T < 33,

where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T(t) = 5t + 1.5, where t is the time in hours. a. Find the composite function N(T(t)). b. Find the time (round to two decimal places) when

the bacteria count reaches 6,752.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

SECTION 3.5 sectioN exercises 243

3.5 SeCTIOn exeRCISeS

veRbAl

1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

2. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

3. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

4. When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?

5. How can you determine whether a function is odd or even from the formula of the function?

AlgebRAIC

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

6. f (x) = √ — x is shifted up 1 unit and to the left 2 units. 7. f (x) = |x| is shifted down 3 units and to the right

1 unit.

8. f (x) =   1 __ x is shifted down 4 units and to the right 3 units.

9. f (x) = 1 __ x2 is shifted up 2 units and to the left 4 units.

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f.

10. y = f (x − 49) 11. y = f (x + 43) 12. y = f (x + 3)

13. y = f (x − 4) 14. y = f (x) + 5 15. y = f (x) + 8

16. y = f (x) − 2 17. y = f (x) − 7 18. y = f (x − 2) + 3

19. y = f (x + 4) − 1

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

20. f (x) = 4(x + 1)2 − 5 21. g(x) = 5(x + 3)2 − 2 22. a(x) = √ —

−x + 4

23. k(x) = −3 √ — x − 1

f2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

Figure 31

gRAPhICAl For the following exercises, use the graph of f (x) = 2x shown in Figure 31 to sketch a graph of each transformation of f (x).

24. g(x) = 2x + 1 25. h(x) = 2x − 3

26. w(x) = 2x − 1

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

27. f (t) = (t + 1)2 − 3 28. h(x) = |x − 1| + 4

29. k(x) = (x − 2)3 − 1 30. m(t) = 3 + √ —

t + 2

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

244 CHAPTER 3 fuNctioNs

nUmeRIC

31. Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f (x).

x –2 –1 0 1 2 f (x) –2 –1 –3 1 2

x –1 0 1 2 3 g(x) –2 –1 –3 1 2

x –2 –1 0 1 2 h(x) –1 0 –2 2 3

32. Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f (x).

x –2 –1 0 1 2 f (x) –1 –3 4 2 1

x –3 –2 –1 0 1 g(x) –1 –3 4 2 1

x –2 –1 0 1 2 h(x) –2 –4 3 1 0

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

33.

f 2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

34.

f

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

35.

f2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

36.

4

x

f

y

–1–2–3–4–5 –2 –4 –6 –8

–10

2

3

6

21 4

8

5

10

37.

4

x

f

y

–1–2–3–4–5 –2 –4 –6 –8

–10

2

3

6

21 4

8

5

10

38.

2

x

f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

 

 

SECTION 3.5 sectioN exercises 245

39.

2

x

f

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

40.

2

x f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

41.

2

x

f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

42.

2

x

f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

43.

2

x f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

44.

2

x

f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

45.

2

x

f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

46.

2

x

f

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

246 CHAPTER 3 fuNctioNs

For the following exercises, determine whether the function is odd, even, or neither. 47. f (x) = 3×4 48. g(x) = √

— x 49. h(x) = 1 __ x + 3x

50. f (x) = (x − 2)2 51. g(x) = 2×4 52. h(x) = 2x − x3

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f.

53. g(x) = −f (x) 54. g(x) = f (−x) 55. g(x) = 4f (x) 56. g(x) = 6f (x)

57. g(x) = f (5x) 58. g(x) = f (2x) 59. g(x) = f   1 __ 3 x  60. g(x) = f   1 __ 5 x 

61. g(x) = 3f (−x) 62. g(x) = −f (3x)

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described.

63. The graph of f (x) = ∣ x ∣ is reflected over the y-axis and horizontally compressed by a factor of 1 __ 4 .

64. The graph of f (x) = √ — x is reflected over the x-axis

and horizontally stretched by a factor of 2.

65. The graph of f (x) = 1 __ x2 is vertically compressed by a factor of 1 __ 3 , then shifted to the left 2 units and down 3 units.

66. The graph of f (x) = 1 __ x is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

67. The graph of f (x) = x2 is vertically compressed by a factor of 1 __ 2 , then shifted to the right 5 units and up 1 unit.

68. The graph of f (x) = x2 is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

69. g(x) = 4(x + 1)2 − 5 70. g(x) = 5(x + 3)2 − 2 71. h(x) = −2 ∣ x − 4 ∣ + 3

72. k(x) = −3 √ — x − 1 73. m(x) =   1 __ 2 x

3 74. n(x) =  1 __ 3 |x − 2|

75. p(x) =      1 __ 3 x  3

− 3 76. q(x) =    1 __ 4 x  3

+ 1 77. a(x) = √ —

−x + 4

For the following exercises, use the graph in Figure 32 to sketch the given transformations.

4

x –2–2–4

–4

–6

–6

–8

–8

–10

–10

2

6

6

42 8

8

10

10

y

f

Figure 32

78. g(x) = f (x) − 2 79. g(x) = −f (x) 80. g(x) = f (x + 1) 81. g(x) = f (x − 2)

 

 

252 CHAPTER 3 fuNctioNs

3.6 SeCTIOn exeRCISeS

veRbAl

1. How do you solve an absolute value equation? 2. How can you tell whether an absolute value function has two x-intercepts without graphing the function?

3. When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

4. How can you use the graph of an absolute value function to determine the x-values for which the function values are negative?

AlgebRAIC

5. Describe all numbers x that are at a distance of 4 from the number 8. Express this using absolute value notation.

6. Describe all numbers x that are at a distance of 1 __ 2 from the number −4. Express this using absolute value notation.

7. Describe the situation in which the distance that point x is from 10 is at least 15 units. Express this using absolute value notation.

8. Find all function values f (x) such that the distance from f (x) to the value 8 is less than 0.03 units. Express this using absolute value notation.

For the following exercises, find the x- and y-intercepts of the graphs of each function.

9. f (x) = 4 ∣ x − 3 ∣ + 4 10. f (x) = −3 ∣ x − 2 ∣ − 1 11. f (x) = −2 ∣ x + 1 ∣ + 6

12. f (x) = −5 ∣ x + 2 ∣ + 15 13. f (x) = 2 ∣ x − 1 ∣ − 6 14. f (x) = ∣ −2x + 1 ∣ − 13

15. f (x) = − ∣ x − 9 ∣ + 16

gRAPhICAl

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

16. y = | x − 1 | 17. y = | x + 1 |  18. y = | x | + 1

For the following exercises, graph the given functions by hand.

19. y = | x | − 2 20. y = −| x |  21. y = −| x | − 2

22. y = −| x − 3 | − 2 23. f (x) = −| x − 1 | − 2 24. f (x) = −| x + 3 | + 4

25. f (x) = 2| x + 3 | + 1 26. f (x) = 3| x − 2 | + 3 27. f (x) = | 2x − 4 | − 3

28. f (x) = | 3x + 9 | + 2 29. f (x) = −| x − 1 | − 3 30. f (x) = −| x + 4 | −3

31. f (x) = 1 __ 2 | x + 4 | − 3

 

 

SECTION 3.6 sectioN exercises 253

TeChnOlOgy

32. Use a graphing utility to graph f (x) = 10| x − 2| on the viewing window [0, 4]. Identify the corresponding range. Show the graph.

33. Use a graphing utility to graph f (x) = −100| x| + 100 on the viewing window [−5, 5]. Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

34. f (x) = −0.1| 0.1(0.2 − x)| + 0.3 35. f (x) = 4 × 109 ∣ x − (5 × 109) ∣  + 2 × 109

exTenSIOnS

For the following exercises, solve the inequality.

36. If possible, find all values of a such that there are no x-intercepts for f (x) = 2| x + 1| + a.

37. If possible, find all values of a such that there are no y-intercepts for f (x) = 2| x + 1| + a.

ReAl-WORld APPlICATIOnS

38. Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and x represents the distance from city B to city A, express this using absolute value notation.

39. The true proportion p of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

40. Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable x for the score.

41. A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x as the diameter of the bearing, write this statement using absolute value notation.

42. The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is x inches, express the tolerance using absolute value notation.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

CHAPTER 3 fuNctioNs264

3.7 SeCTIOn exeRCISeS

veRbAl

1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

2. Why do we restrict the domain of the function f (x) = x2 to find the function’s inverse?

3. Can a function be its own inverse? Explain. 4. Are one-to-one functions either always increasing or always decreasing? Why or why not?

5. How do you find the inverse of a function algebraically?

AlgebRAIC

6. Show that the function f (x) = a − x is its own inverse for all real numbers a.

For the following exercises, find f −1(x) for each function.

7. f (x) = x + 3 8. f (x) = x + 5 9. f (x) = 2 − x

10. f (x) = 3 − x 11. f (x) = x _____ x + 2 12. f (x) = 2x + 3 ______ 5x + 4

For the following exercises, find a domain on which each function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

13. f (x) = (x + 7)2 14. f (x) = (x − 6)2 15. f (x) = x 2 − 5

16. Given f(x) = x _____ 2 + x and g(x) = 2x _____ 1 − x :

a. Find f (g(x)) and g (f (x)). b. What does the answer tell us about the relationship between f (x) and g(x)?

For the following exercises, use function composition to verify that f (x) and g(x) are inverse functions.

17. f (x) = 3 √ —

x − 1 and g (x) = x 3 + 1 18. f (x) = −3x + 5 and g (x) = x − 5 _____ −3

gRAPhICAl

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

19. f (x) = √ — x 20. f (x) = 3 √

— 3x + 1

21. f (x) = −5x + 1 22. f (x) = x 3 − 27

For the following exercises, determine whether the graph represents a one-to-one function.

23.

10

x

y

f

–5–5–10

–10

–15

–15

–20

–20

–25

–25

5

15

15

105 20

20

25

25

24.

4

x

y

f –2–4–6–8–10 –2

–4 –6 –8

–10

2

6

6

42 8

8

10

10

 

 

SECTION 3.7 sectioN exercises 265

For the following exercises, use the graph of f shown in Figure 11.

2

x

y

f

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

Figure 11

25. Find f (0).

26. Solve f (x) = 0.

27. Find f −1(0).

28. Solve f −1(x) = 0.

For the following exercises, use the graph of the one-to-one function shown in Figure 12.

4

x

y

f

–2–4–6–8–10 –2 –4 –6 –8

–10

2

6

6

42 8

8

10

10

Figure 12

29. Sketch the graph of f −1.

30. Find f (6) and f −1(2).

31. If the complete graph of f is shown, find the domain of f.

32. If the complete graph of f is shown, find the range of f.

nUmeRIC

For the following exercises, evaluate or solve, assuming that the function f is one-to-one.

33. If f (6) = 7, find f −1(7). 34. If f (3) = 2, find f −1(2).

35. If f −1(−4) = −8, find f (−8). 36. If f −1(−2) = −1, find f (−1).

For the following exercises, use the values listed in Table 6 to evaluate or solve.

x 0 1 2 3 4 5 6 7 8 9

f (x) 8 0 7 4 2 6 5 3 9 1

Table 6

37. Find f (1). 38. Solve f (x) = 3. 39. Find f −1(0). 40. Solve f −1(x) = 7.

41. Use the tabular representation of f in Table 7 to create a table for f −1 (x).

x 3 6 9 13 14

f (x) 1 4 7 12 16

Table 7

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

266 CHAPTER 3 fuNctioNs

TeChnOlOgy

For the following exercises, find the inverse function. Then, graph the function and its inverse.

42. f (x) = 3 _____ x − 2 43. f (x) = x 3 − 1

44. Find the inverse function of f (x) = 1 _ x − 1

. Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

ReAl-WORld APPlICATIOnS

45. To convert from x degrees Celsius to y degrees Fahrenheit, we use the formula f (x) = 9 __ 5 x + 32. Find the inverse function, if it exists, and explain its meaning.

46. The circumference C of a circle is a function of its radius given by C(r) = 2πr. Express the radius of a circle as a function of its circumference. Call this function r(C). Find r(36π) and interpret its meaning.

47. A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, t, in hours given by d(t) = 50t. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function t(d). Find t(180) and interpret its meaning.

 

 

CHAPTER 3 review 267

ChAPTeR 3 RevIeW

Key Terms absolute maximum the greatest value of a function over an interval

absolute minimum the lowest value of a function over an interval

average rate of change the difference in the output values of a function found for two values of the input divided by the difference between the inputs

composite function the new function formed by function composition, when the output of one function is used as the input of another

decreasing function a function is decreasing in some open interval if f (b) < f (a) for any two input values a and b in the given interval where b > a

dependent variable an output variable

domain the set of all possible input values for a relation

even function a function whose graph is unchanged by horizontal reflection, f (x) = f (−x), and is symmetric about the y-axis

function a relation in which each input value yields a unique output value

horizontal compression a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant b > 1

horizontal line test a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once

horizontal reflection a transformation that reflects a function’s graph across the y-axis by multiplying the input by −1

horizontal shift a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input

horizontal stretch a transformation that stretches a function’s graph horizontally by multiplying the input by a constant 0 < b < 1

increasing function a function is increasing in some open interval if f (b) > f (a) for any two input values a and b in the given interval where b > a

independent variable an input variable

input each object or value in a domain that relates to another object or value by a relationship known as a function

interval notation a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion

inverse function for any one-to-one function f (x), the inverse is a function f −1(x) such that f −1(f (x)) = x for all x in the domain of f; this also implies that f ( f −1(x)) = x for all x in the domain of f −1

local extrema collectively, all of a function’s local maxima and minima

local maximum a value of the input where a function changes from increasing to decreasing as the input value increases.

local minimum a value of the input where a function changes from decreasing to increasing as the input value increases.

odd function a function whose graph is unchanged by combined horizontal and vertical reflection, f (x) = − f (−x), and is symmetric about the origin

one-to-one function a function for which each value of the output is associated with a unique input value

output each object or value in the range that is produced when an input value is entered into a function

piecewise function a function in which more than one formula is used to define the output

range the set of output values that result from the input values in a relation

rate of change the change of an output quantity relative to the change of the input quantity

relation a set of ordered pairs

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

268 CHAPTER 3 fuNctioNs

set-builder notation a method of describing a set by a rule that all of its members obey; it takes the form {x ∣ statement about x}

vertical compression a function transformation that compresses the function’s graph vertically by multiplying the output by a constant 0 < a < 1

vertical line test a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once

vertical reflection a transformation that reflects a function’s graph across the x-axis by multiplying the output by −1

vertical shift a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output

vertical stretch a transformation that stretches a function’s graph vertically by multiplying the output by a constant a > 1

Key equations

Constant function f (x) = c, where c is a constant

Identity function f (x) = x

Absolute value function f (x) = ∣x∣

Quadratic function f (x) = x2

Cubic function f (x) = x3

Reciprocal function f (x) = 1 _ x

Reciprocal squared function f (x) = 1 _ x2

 

Square root function f (x) = √ — x

Cube root function f (x) = 3 √ — x

Average rate of change ∆y

_ ∆x

= f (x2) − f (x1) _ x2 − x1

 

Composite function (f ∘ g)(x) = f (g(x))

Vertical shift g(x) = f (x) + k (up for k > 0)

Horizontal shift g(x) = f (x − h) (right for h > 0)

Vertical reflection g(x) = −f (x)

Horizontal reflection g(x) = f (−x)

Vertical stretch g(x) = af (x) (a > 0)

Vertical compression g(x) = af (x) (0 < a < 1)

Horizontal stretch g(x) = f (bx) (0 < b < 1)

Horizontal compression g(x) = f (bx) (b > 1)

 

 

CHAPTER 3 review 269

Key Concepts

3.1 Functions and Function Notation • A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input,

leads to exactly one range value, or output. See Example 1 and Example 2.

• Function notation is a shorthand method for relating the input to the output in the form y = f (x). See Example 3 and Example 4.

• In tabular form, a function can be represented by rows or columns that relate to input and output values. See Example 5.

• To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See Example 6 and Example 7.

• To solve for a specific function value, we determine the input values that yield the specific output value. See Example 8.

• An algebraic form of a function can be written from an equation. See Example 9 and Example 10.

• Input and output values of a function can be identified from a table. See Example 11.

• Relating input values to output values on a graph is another way to evaluate a function. See Example 12.

• A function is one-to-one if each output value corresponds to only one input value. See Example 13.

• A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See Example 14.

• The graph of a one-to-one function passes the horizontal line test. See Example 15.

3.2 Domain and Range • The domain of a function includes all real input values that would not cause us to attempt an undefined

mathematical operation, such as dividing by zero or taking the square root of a negative number.

• The domain of a function can be determined by listing the input values of a set of ordered pairs. See Example 1.

• The domain of a function can also be determined by identifying the input values of a function written as an equation. See Example 2, Example 3, and Example 4.

• Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See Example 5.

• For many functions, the domain and range can be determined from a graph. See Example 6 and Example 7.

• An understanding of toolkit functions can be used to find the domain and range of related functions. See Example 8, Example 9, and Example 10.

• A piecewise function is described by more than one formula. See Example 11 and Example 12.

• A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See Example 13.

3.3 Rates of Change and Behavior of Graphs • A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is

determined using only the beginning and ending data. See Example 1.

• Identifying points that mark the interval on a graph can be used to find the average rate of change. See Example 2.

• Comparing pairs of input and output values in a table can also be used to find the average rate of change. See Example 3.

• An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See Example 4 and Example 5.

• The average rate of change can sometimes be determined as an expression. See Example 6.

• A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See Example 7.

• A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

270 CHAPTER 3 fuNctioNs

• A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.

• Minima and maxima are also called extrema.

• We can find local extrema from a graph. See Example 8 and Example 9.

• The highest and lowest points on a graph indicate the maxima and minima. See Example 10.

3.4 Composition of Functions • We can perform algebraic operations on functions. See Example 1.

• When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.

• The function produced by combining two functions is a composite function. See Example 2 and Example 3.

• The order of function composition must be considered when interpreting the meaning of composite functions. See Example 4.

• A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.

• A composite function can be evaluated from a table. See Example 5.

• A composite function can be evaluated from a graph. See Example 6.

• A composite function can be evaluated from a formula. See Example 7.

• The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See Example 8 and Example 9.

• Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.

• Functions can often be decomposed in more than one way. See Example 10.

3.5 Transformation of Functions • A function can be shifted vertically by adding a constant to the output. See Example 1 and Example 2.

• A function can be shifted horizontally by adding a constant to the input. See Example 3, Example 4, and Example 5.

• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See Example 6.

• Vertical and horizontal shifts are often combined. See Example 7 and Example 8.

• A vertical reflection reflects a graph about the x-axis. A graph can be reflected vertically by multiplying the output by –1.

• A horizontal reflection reflects a graph about the y-axis. A graph can be reflected horizontally by multiplying the input by –1.

• A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See Example 9.

• A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See Example 10.

• A function presented as an equation can be reflected by applying transformations one at a time. See Example 11.

• Even functions are symmetric about the y-axis, whereas odd functions are symmetric about the origin.

• Even functions satisfy the condition f (x) = f (−x).

• Odd functions satisfy the condition f (x) = −f (−x).

• A function can be odd, even, or neither. See Example 12.

• A function can be compressed or stretched vertically by multiplying the output by a constant. See Example 13, Example 14, and Example 15.

• A function can be compressed or stretched horizontally by multiplying the input by a constant. See Example 16, Example 17, and Example 18.

 

 

CHAPTER 3 review 271

• The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See Example 19 and Example 20.

3.6 Absolute Value Functions • Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See

Example 1.

• The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See Example 2.

• In an absolute value equation, an unknown variable is the input of an absolute value function.

• If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See Example 3.

3.7 Inverse Functions • If g(x) is the inverse of f (x), then g(f (x)) = f (g(x)) = x. See Example 1, Example 2, and Example 3.

• Each of the toolkit functions has an inverse. See Example 4.

• For a function to have an inverse, it must be one-to-one (pass the horizontal line test).

• A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.

• For a tabular function, exchange the input and output rows to obtain the inverse. See Example 5.

• The inverse of a function can be determined at specific points on its graph. See Example 6.

• To find the inverse of a formula, solve the equation y = f (x) for x as a function of y. Then exchange the labels x and y. See Example 7, Example 8, and Example 9.

• The graph of an inverse function is the reflection of the graph of the original function across the line y = x. See Example 10.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

272 CHAPTER 3 fuNctioNs

ChAPTeR 3 RevIeW exeRCISeS

FUnCTIOnS And FUnCTIOn nOTATIOn

For the following exercises, determine whether the relation is a function.

1. {(a, b), (c, d), (e, d)}

2. {(5, 2), (6, 1), (6, 2), (4, 8)}

3. y 2 + 4 = x, for x the independent variable and y the dependent variable

4. Is the graph in Figure 1 a function? y

–5–10–15–20–25 –5 –10 –15 –20

5

15 20

Figure 1

For the following exercises, evaluate the function at the indicated values: f (−3); f (2); f (−a); −f (a); f (a + h).

5. f (x) = −2x 2 + 3x 6. f(x) = 2∣3x − 1∣

For the following exercises, determine whether the fun ctions are one-to-one.

7. f (x) = −3x + 5 8. f (x) = ∣x − 3∣

For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

9.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

10.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

11.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

For the following exercises, graph the functions.

12. f (x) = ∣x + 1∣ 13. f (x) = x 2 − 2

5 10 15 20 25

10

 

 

CHAPTER 3 review 273

For the following exercises, use Figure 2 to approximate the values.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

Figure 2

14. f (2)

15. f (−2)

16. If f (x) = −2, then solve for x.

17. If f (x) = 1, then solve for x.

For the following exercises, use the function h(t) = −16t 2 + 80t to find the values.

18. h(2) − h(1)

__2 − 1 19. h(a) − h(1)

__a − 1

dOmAIn And RAnge

For the following exercises, find the domain of each function, expressing answers using interval notation.

20. f (x) = 2 _3x + 2 21. f (x) = x − 3 ___________x 2 − 4x − 12 22. f (x) =

√ — x − 6 _

√ —

x − 4

23. Graph this piecewise function: f (x) = { x + 1 x < −2 −2x − 3 x ≥ −2

RATeS OF ChAnge And behAvIOR OF gRAPhS

For the following exercises, find the average rate of change of the functions from x = 1 to x = 2.

24. f (x) = 4x − 3 25. f (x) = 10x 2 + x 26. f (x) = −  2 _ x 2

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

27.

4

x

y

–1–2–3–4–5 –2 –4 –6 –8

–10

2

3

6

21 4

8

5

10

28.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

29.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

30. Find the local minimum of the function graphed in Exercise 27.

31. Find the local extrema for the function graphed in Exercise 28.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

274 CHAPTER 3 fuNctioNs

32. For the graph in Figure 3, the domain of the function is [−3, 3]. The range is [−10, 10]. Find the absolute minimum of the function on this interval.

4

x

y

–1–2–3–4–5 –2 –4 –6 –8

–10

2

3

6

21 4

8

5

10

Figure 3

33. Find the absolute maximum of the function graphed in Figure 3.

COmPOSITIOn OF FUnCTIOnS

For the following exercises, find (f ∘ g)(x) and (g ∘ f)(x) for each pair of functions.

34. f (x) = 4 − x, g(x) = −4x 35. f (x) = 3x + 2, g(x) = 5 − 6x 36. f (x) = x 2 + 2x, g(x) = 5x + 1

37. f (x) = √ —

x + 2 , g(x) = 1 _ x 38. f (x) = x + 3 _____ 2 , g(x) = √

— 1 − x

For the following exercises, find (f ∘ g) and the domain for (f ∘ g)(x) for each pair of functions.

39. f (x) = x + 1 _ x + 4 , g(x) = 1 __ x 40. f (x) =

1 _ x + 3 , g(x) = 1 _ x − 9 41. f (x) =

1 _ x , g(x) = √ — x

42. f (x) = 1 _ x2 − 1

, g(x) = √ —

x + 1

For the following exercises, express each function H as a composition of two functions f and g where H(x) = (f ∘ g)(x).

43. H(x) = √ _______

2x − 1 ______3x + 4 44. H(x) = 1 _

(3×2 − 4)−3

TRAnSFORmATIOn OF FUnCTIOnS

For the following exercises, sketch a graph of the given function.

45. f (x) = (x − 3)2 46. f (x) = (x + 4)3 47. f (x) = √ — x + 5

48. f (x) = −x3 49. f (x) = 3 √ — −x 50. f (x) = 5 √

— −x − 4

51. f (x) = 4[∣x − 2∣ − 6] 52. f (x) = −(x + 2)2 − 1

For the following exercises, sketch the graph of the function g if the graph of the function f is shown in Figure 4.

2

x

y

–1–2–3–4–5 –1 –2

1

3

3

21 4

4

5

5

Figure 4

53. g(x) = f (x − 1)

54. g(x) = 3f (x)

 

 

CHAPTER 3 review 275

For the following exercises, write the equation for the standard function represented by each of the graphs below.

55.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

56.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

For the following exercises, determine whether each function below is even, odd, or neither.

57. f (x) = 3×4 58. g(x) = √ — x 59. h(x) = 1 _ x + 3x

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

60.

10

x

y

–5–10–15–20–25 –5 –10 –15 –20 –25

5

15

15

105 20

20

25

25

61.

10

x

y

–2–4–6–8–10 –5 –10 –15 –20 –25

5

6

15

42 8

20

10

25

62.

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10

2

6

6

42 8

8

10

10

AbSOlUTe vAlUe FUnCTIOnS

For the following exercises, write an equation for the transformation of f (x) = | x |.

63.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

64.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

65.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

For the following exercises, graph the absolute value function.

66. f (x) = | x − 5 | 67. f (x) = −| x − 3 | 68. f (x) = | 2x − 4 |

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

276 CHAPTER 3 fuNctioNs

InveRSe FUnCTIOnS

For the following exercises, find f −1(x) for each function.

69. f (x) = 9 + 10x 70. f (x) = x _ x + 2

For the following exercise, find a domain on which the function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

71. f (x) = x 2 + 1

72. Given f (x) = x 3 − 5 and g(x) = 3 √ —

x + 5 : a. Find f (g(x)) and g(f (x)). b. What does the answer tell us about the relationship between f (x) and g(x)?

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

73. f (x) = 1 _ x 74. f (x) = −3x 2 + x 75. If f (5) = 2, find f −1(2).

76. If f (1) = 4, find f −1(4).

 

 

277CHAPTER 3 Practice test

ChAPTeR 3 PRACTICe TeST

For the following exercises, determine whether each of the following relations is a function.

1. y = 2x + 8 2. {(2, 1), (3, 2), (−1, 1), (0, −2)}

For the following exercises, evaluate the function f (x) = −3x 2 + 2x at the given input.

3. f (−2) 4. f (a) 5. Show that the function f (x) = −2(x − 1)2 + 3 is not

one-to-one. 6. Write the domain of the function f (x) = √

— 3 − x in

interval notation.

7. Given f (x) = 2x 2 − 5x, find f (a + 1) − f (1). 8. Graph the function f (x) = { x + 1 if −2 < x < 3−x if x ≥ 3

9. Find the average rate of change of the function f (x) = 3 − 2x 2 + x by finding f (b) − f (a)

_ b − a

.

For the following exercises, use the functions f (x) = 3 − 2x 2 + x and g(x) = √ — x to find the composite functions.

10. ( g ∘ f )(x) 11. ( g ∘ f )(1)

12. Express H(x) = 3 √ —

5x 2 − 3x as a composition of two functions, f and g, where ( f ∘ g )(x) = H(x).

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

13. f (x) = √ —

x + 6 − 1 14. f (x) = 1 _ x + 2 − 1

For the following exercises, determine whether the functions are even, odd, or neither.

15. f (x) = − 5 _ x2

+ 9x 6 16. f (x) = − 5 _ x 3

+ 9x 5

17. f (x) = 1 _ x 18. Graph the absolute value function

f (x) = −2| x − 1 | + 3.

For the following exercises, find the inverse of the function.

19. f (x) = 3x − 5 20. f (x) = 4 _____ x + 7

For the following exercises, use the graph of g shown in Figure 1.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

Figure 1

21. On what intervals is the function increasing?

22. On what intervals is the function decreasing?

23. Approximate the local minimum of the function. Express the answer as an ordered pair.

24. Approximate the local maximum of the function. Express the answer as an ordered pair.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

278 CHAPTER 3 fuNctioNs

For the following exercises, use the graph of the piecewise function shown in Figure 2.

2

x

y

f

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

Figure 2

25. Find f (2).

26. Find f (−2).

27. Write an equation for the piecewise function.

For the following exercises, use the values listed in Table 1.

x 0 1 2 3 4 5 6 7 8 F (x) 1 3 5 7 9 11 13 15 17

Table 1

28. Find F (6). 29. Solve the equation F (x) = 5.

30. Is the graph increasing or decreasing on its domain? 31. Is the function represented by the graph one-to-one?

32. Find F −1(15). 33. Given f (x) = −2x + 11, find f −1(x).

 

 

ODD ANSWERS C-5

Chapter 2 practice test 1. y = 3 _ 2 x + 2

x 0 2 4 y 2 5 8

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

3. (0, −3) (4, 0)

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4 5

5

(−1, 2) (2, 1)

(0, 4)

5. (−∞, 9] 7. x = −15 9. x ≠ −4, 2; x = − 5 _ 2 , 1

11. x = 3 ± √ — 3 _______

2 13. (−4, 1) 15. y = − 5 _ 9 x −

2 _ 9

17. y = 5 _ 2 x − 4 19. 14i 21. 5 ___

13 − 14 ___

13 i 23. x = 2, − 4 _ 3

25. x = 1 _ 2 ± √

— 2 _ 2 27. 4 29. x =

1 _ 2 , 2, −2

ChapteR 3

Section 3.1 1. A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate. 3. When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function. 5. When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input. 7. Function 9. Function 11. Function 13. Function 15. Function 17. Function 19. Function 21. Function 23. Function 25. Not a function 27. f (−3) = −11, f (2) = −1, f (−a) = −2a − 5, −f (a) = −2a + 5, f (a + h) = 2a + 2h − 5 29. f (−3) = √

— 5 + 5, f (2) = 5,

f (−a) = √ —

2 + a + 5, −f (a) = − √ —

2 − a − 5, f (a + h) = √ —

2 − a − h + 5 31. f (−3) = 2, f (2) = −2, f (−a) = ∣ −a − 1 ∣ − ∣ −a + 1 ∣ , −f (a) = − ∣ a − 1 ∣ + ∣ a + 1 ∣ , f (a + h) = ∣ a + h − 1 ∣ − ∣ a + h + 1 ∣ 33.

g(x) − g(a) _ x − a = x + a + 2, x ≠ a 35. a. f (−2) = 14 b. x = 3

37. a. f (5) = 10 b. x = 4 or −1 39. a. r = 6 − 2 __ 3 t

b. f (−3) = 8 c. t = 6 41. Not a function 43. Function 45. Function 47. Function 49. Function 51. Function 53. a. f (0) = 1 b. f (x) = −3, x = −2 or 2 55. Not a function, not one-to-one 57. One-to-one function 59. Function, not one-to-one 61. Function 63. Function 65. Not a function 67. f (x) = 1, x = 2 69. f (−2) = 14; f (−1) = 11; f (0) = 8; f (1) = 5; f (2) = 2

71. f (−2) = 4; f (−1) = 4.414; f (0) = 4.732; f (1) = 5; f (2) = 5.236

73. f (−2) = 1 __ 9 ; f (−1) = 1 __ 3 ; f (0) = 1; f (1) = 3; f (2) = 9 75. 20

77. The range for this viewing window is [0, 100].

x

y

20

–20 –40 –60 –80

–100

40 60 80

100

–5 5–10 10

79. The range for this viewing window is [−0.001, 0.001].

x

y

0.0002

–0.0002 –0.0004 –0.0006 –0.0008 –0.001

0.0004 0.0006 0.0008

0.001

–0.05–0.1 0.05 0.1

81. The range for this viewing window is [−1,000,000, 1,000,000].

x

y

2.105

–2.105 –4.105 –6.105 –8.105

–10.105

4.105 6.105 8.105

10.105

–50–100 50 100

83. The range for this viewing window is [0, 10].

10 8 6 4 2

–20 x

y

20 40 60 80 100

85. The range for this viewing window is [−0.1, 0.1].

x

y

0.02 0.04 0.06 0.08

0.1

–0.02 –0.0005–0.0001 0.0001

–0.04 –0.06 –0.08 –0.1

0.0005

87. The range for this viewing window is [−100, 100].

x

y

20 40 60 80

100

–20–5 .105–10.105 5.105 10.105

–40 –60 –80

–100

89. a. g(5000) = 50 b. The number of cubic yards of dirt required for a garden of 100 square feet is 1. 91. a. The height of the rocket above ground after 1 second is 200 ft. b. The height of the rocket above ground after 2 seconds is 350 ft.

Section 3.2 1. The domain of a function depends upon what values of the independent variable make the function undefined or imaginary. 3. There is no restriction on x for f (x) = 3 √

— x because you can

take the cube root of any real number. So the domain is all real numbers, (−∞, ∞). When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x-values are restricted for f (x) = √

— x to nonnegative numbers and the

domain is [0, ∞). 5. Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x-axis and y-axis for each graph. Indicate included endpoints with a solid circle and excluded endpoints with an open circle. Use an arrow to indicate −∞ or ∞. Combine the graphs to find the graph of the piecewise function. 7. (−∞, ∞) 9. (−∞, 3]

11. (−∞, ∞) 13. (−∞, ∞) 15.  −∞, − 1 _ 2  ∪  − 1 _ 2 , ∞ 

17. (−∞, −11)∪(−11, 2)∪(2, ∞) 19. (−∞, −3)∪(−3, 5)∪(5, ∞)

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

ODD ANSWERSC-6

21. (−∞, 5) 23. [6, ∞) 25. (−∞, −9)∪(−9, 9)∪(9, ∞) 27. Domain: (2, 8], range: [6, 8) 29. Domain: [−4, 4], range: [0, 2] 31. Domain: [−5, 3), range: [0, 2] 33. Domain: (−∞, 1], range: [0, ∞)

35. Domain:  −6, − 1 _ 6  ∪  1 _ 6

, 6  , range:  −6, − 1 _ 6  ∪  1 _ 6

, 6  37. Domain: [−3, ∞), range is [0, ∞) 39. Domain:(−∞, ∞)

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4 5

5

41. Domain: (−∞, ∞)

x

y

1 2

–1–2

–2 –3

–1

–4 –5

1 2

4 5

3

43. Domain: (−∞, ∞)

1 2

x

y

–1–2

–2 –3

–1

–4 –5

1 2

4 5

3

45. Domain: (−∞, ∞)

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

x

y

47. f (−3) = 1; f (−2) = 0; f (−1) = 0; f (0) = 0 49. f (−1) = −4; f (0) = 6; f (2) = 20; f (4) = 34 51. f (−1) = −5; f (0) = 3; f (2) = 3; f (4) = 16 53. (−∞, 1)∪(1, ∞) 55.

16 24

x

y

8

–8

32

0.1

40 48 56 64 72 80 88 96

104

–0.1–0.2–0.3–0.4–0.5

The viewing window: [−0.5, −0.1] has a range: [4, 100].

 

16 24

y

x 8

–8–0.1

32 40 48 56 64 72 80 88 96

104

0.1 0.2 0.3 0.4 0.5

The viewing window: [0.1, 0.5] has a range: [4, 100].

57. [0, 8] 59. Many answers; one function is f (x) = 1 _______ √ —

x − 2 .

61. a. The fixed cost is $500. b. The cost of making 25 items is $750. c. The domain is [0, 100] and the range is [500, 1500].

Section 3.3 1. Yes, the average rate of change of all linear functions is constant. 3. The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region in an open interval. 5. 4(b + 1) 7. 3 9. 4x + 2h 11. −1 _________ 13(13 + h) 13. 3h

2 + 9h + 9 15. 4x + 2h − 3

17. 4 _ 3 19. Increasing on (−∞, −2.5)∪(1, ∞) a nd decreasing

on (−2.5, 1) 21. Increasing on (−∞, 1)∪(3, 4) and decreasing on (1, 3)∪(4, ∞) 23. Local maximum: (−3, 50) and local

minimum: (3, 50) 25. Absolute maximum at approximately (7, 150) and absolute minimum at approximately (−7.5, −220) 27. a. −3,000 people per year b. −1,250 people per year 29. −4 31. 27 33. ≈ −0.167 35. Local minimum: (3, −22), decreasing on (−∞, 3), increasing on (3, ∞) 37. Local minimum: (−2, −2), decreasing on (−3, −2), increasing on (−2, ∞) 39. Local maximum: (−0.5, 6), local minima: (−3.25, −47) and (2.1, −32), decreasing on (−∞, −3.25) and (−0.5, 2.1), increasing on (−3.25, −0.5) and (2.1, ∞) 41. A 43. b = 5 45. ≈ 2.7 gallons per minute 47. ≈ −0.6 milligrams per day

Section 3.4 1. Find the numbers that make the function in the denominator g equal to zero, and check for any other domain restrictions on f and g, such as an even-indexed root or zeros in the denominator. 3. Yes, sample answer: Let f (x) = x + 1 and g(x) = x − 1. Then f (g(x)) = f (x − 1) = (x − 1)+ 1 = x and g ( f (x)) = g (x + 1) = (x + 1)− 1 = x so f ∘ g = g ∘ f. 5. (f + g)(x) = 2x + 6; domain: (−∞, ∞) (f − g)(x) = 2x 2 + 2x − 6; domain: (−∞, ∞) (fg)(x) = −x 4 − 2x 3 + 6x 2 + 12x; domain: (−∞, ∞)

 f _ g  (x) = x 2 + 2x ______ 6 − x 2 ; domain: (−∞, − √

— 6 )∪(− √

— 6 , √

— 6 )∪( √

— 6 , ∞)

7. (f + g)(x) = 4x 3 + 8×2 + 1 ___________ 2x ; domain: (−∞, 0)∪(0, ∞)

(f − g)(x) = 4x 3 + 8×2 − 1 ___________ 2x ; domain: (−∞, 0)∪(0, ∞)

(fg)(x) = x + 2; domain: (−∞, 0)∪(0, ∞)

 f _ g  (x) = 4x 3 + 8x 2; domain: (−∞, 0)∪(0, ∞) 9. (f + g)(x) = 3x 2 + √

— x − 5 ; domain: [5, ∞)

(f − g)(x) = 3x 2 − √ —

x − 5 ; domain: [5, ∞) (fg)(x) = 3x 2 √

— x − 5 ; domain: [5, ∞)

 f _ g  (x) = 3x 2 _

√ —

x − 5 ; domain: (5, ∞) 11. a. f (g(2)) = 3

b. f ( g(x)) = 18x 2 − 60x + 51 c. g( f (x)) = 6x 2 − 2 d. ( g ∘ g)(x) = 9x − 20 e. ( f ∘ f )(−2) = 163

13. f ( g(x)) = √ —

x2 + 3 + 2 ; g( f (x)) = x + 4 √ — x + 7

15. f ( g(x)) = 3 √ —

x + 1 _ x ; g( f (x)) = 3 √

— x + 1 _ x

17. f ( g(x)) = x __ 2 , x ≠ 0; g( f (x)) = 2x − 4, x ≠ 4

19. f ( g(h(x))) = 1 _ (x + 3)2

+ 1

21. a. (g ∘ f )(x) = − 3 _ √ —

2 − 4x b.  −∞, 1 __ 2 

23. a. (0, 2)∪(2, ∞) except x = −2 b. (0, ∞) c. (0, ∞) 25. (1, ∞)

27. Many solutions; one possible answer: f (x) = x3; g(x) = x − 5

29. Many solutions; one possible answer: f (x) = 4 _ x ; g(x) = (x + 2) 2

31. Many solutions; one possible answer: f (x) = 3 √ — x ; g(x) = 1 _____ 2x − 3

33. Many solutions; one possible answer: f (x) = 4 √ — x ; g(x) = 3x − 2 ______ x + 5

35. Many solutions; one possible answer: f (x) = √ — x ; g(x) = 2x + 6

37. Many solutions; one possible answer: f (x) = 3 √ — x ; g(x) = x − 1

39. Many solutions; one possible answer: f (x) = x 3; g(x) = 1 _ x − 2

 

 

ODD ANSWERS C-7

41. Many solutions; one possible answer: f (x) = √ — x ; g(x) = 2x − 1 _ 3x + 4

43. 2 45. 5 47. 4 49. 0 51. 2 53. 1 55. 4 57. 4 59. 9 61. 4 63. 2 65. 3 67. 11 69. 0 71. 7 73. f (g(0)) = 27, g(f (0)) = −94

75. f (g(0)) = 1 _ 5 , g(f (0)) = 5 77. f (g(x)) = 18x 2 + 60x + 51

79. g ∘ g(x) = 9x + 20 81. ( f ∘ g)(x) = 2, (g ∘ f )(x) = 2 83. (−∞, ∞) 85. False 87. (f ∘ g )(6) = 6; (g ∘ f )(6) = 6 89. ( f ∘ g )(11) = 11; (g ∘ f )(11) = 11 91. C

93. A(t) = π  25 √ —

t + 2  2 and A(2) = π  25 √ — 4  2 = 2,500π

square inches 95. A(5) = 121π square units 97. a. N(T(t)) = 575t 2 + 65t − 31.25 b. ≈ 3.38 hours

Section 3.5 1. A horizontal shift results when a constant is added to or subtracted from the input. A vertical shift results when a constant is added to or subtracted from the output. 3. A horizontal compression results when a constant greater than 1 multiplies the input. A vertical compression results when a constant between 0 and 1 multiplies the output. 5. For a function f, substitute (−x) for (x) in f (x) and simplify. If the resulting function is the same as the original function, f (−x) = f (x), then the function is even. If the resulting function is the opposite of the original function, f (−x) = −f (x), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even. 7. g(x) = ∣ x − 1 ∣ − 3 9. g(x) = 1 _

(x + 4)2 + 2 11. The graph of f (x + 43) is a

horizontal shift to the left 43 units of the graph of f. 13. The graph of f (x − 4) is a horizontal shift to the right 4 units of the graph of f. 15. The graph of f (x) + 8 is a vertical shift up 8 units of the graph of f. 17. The graph of f (x) − 7 is a vertical shift down 7 units of the graph of f. 19. The graph of f (x + 4) − 1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f. 21. Decreasing on (−∞, −3) and increasing on (−3, ∞) 23. Decreasing on (0, ∞) 25.

h

2

x

y

–1–2–3–4–5–6–7–8–9 –1 –2 –3 –4

1

0 3

3

21 4

4 5 6 7 8 9

10

27.

f

2

x

y

–1–2–3–4–5 –1 –2 –3 –4

1

3

3

21

4 5 6 7 8 9

10

29. 31. g(x) = f (x − 1), h(x) = f (x) + 1 33. f (x) = ∣ x − 3 ∣ − 2 35. f (x) = √

— x + 3 − 1

37. f (x) = (x − 2)2 39. f (x) = ∣ x + 3 ∣ − 2 41. f (x) = − √

— x

43. f (x) = −(x + 1)2 + 2 45. f (x) = √

— −x + 1

47. Even 49. Odd 51. Even 53. The graph of g is a vertical reflection (across the x-axis) of the graph of f. 55. The graph of g is a vertical stretch by a factor of 4 of the graph of f.

 

k

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6 –7 –8 –9

–10 –11 –12 –13 –14

1

3

3

21 4

4

5 6

5 6 7 8 9

57. The graph of g is a horizontal compression by a factor of 1 _ 5

of the graph of f. 59. The graph of g is a horizontal stretch by a factor of 3 of the graph of f. 61. The graph of g is a horizontal reflection across the y-axis and a vertical stretch by a

factor of 3 of the graph of f. 63. g(x) = ∣ −4x ∣ 65. g(x) = 1 _

3(x + 2)2 − 3 67. g(x) = 1 _ 2 (x − 5)

2 + 1

69. This is a parabola shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

g

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5 6 7 8 9

10

71. This is an absolute value function stretched vertically by a factor of 2, shifted 4 units to the right, reflected across the horizontal axis, and then shifted 3 units up.

h 2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6 –7 –8 –9

–10

1

3

3

21 4

4

5 6

73. This is a cubic function compressed vertically by a factor of 1 _ 2 .

m 2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5 –6 –7 –8

1

3

3

21 4

4

5

5 6 7 8

75. The graph of the function is stretched horizontally by a factor of 3 and then shifted downward by 3 units.

p

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6 –7 –8

1

3

3

21 4

4

5 6 7

5 6 7 8

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

 

 

ODD ANSWERSC-8

77. The graph of f (x) = √

— x is shifted

right 4 units and then reflected across the y-axis.

0.2 0 0.5 1.5 2.5 3.52 3 41

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.2 1.3 1.4 1.5 1.6

a

1.7 1.8 1.9

2

1

y

x

79.

g2

x

y

–1–2–3–4–5–6–7–8 –1 –2 –3 –4 –5

1

3

3

21 4

4 5

5 6 7 8

81.

2

x

y

–1–2–3–4–5–6–7–8 –1 –2 –3 –4 –5 –6 –7 –8

1

3

3

21 4

4

5 6 7 8

g

Section 3.6 1. Isolate the absolute value term so that the equation is of the form ∣ A ∣ = B. Form one equation by setting the expression inside the absolute value symbol, A, equal to the expression on the other side of the equation, B. Form a second equation by setting A equal to the opposite of the expression on the other side of the equation, −B. Solve each equation for the variable. 3. The graph of the absolute value function does not cross the x-axis, so the graph is either completely above or completely below the x-axis. 5. The distance from x to 8 can be represented using the absolute value statement: ∣ x − 8 ∣ = 4. 7. ∣ x − 10 ∣ ≥ 15 9. There are no x-intercepts. 11. (−4, 0) and (2, 0)

13. (−2, 0), (4, 0), and (0, −4) 15. (−7, 0), (0, 16), (25, 0)

17.

(–1, 0)

(0, 1) (1, 2)

(–2, 1) (–3, 2)

x

y 19.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

21.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5 –6 –7

1

3

3

21 4

4

5

5

23.

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

25.

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

27.

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5

1

3

3

21 4

4

5 6

5

29.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5

5 6

31.

x y

–1–2–3–4–5–6–7 –1 –2 –3 –4 –5 –6

321 4 5

33. range: [−400, 100]

x

y

–1–2–3–4–5

50

–50 –100 –150 –200 –250

–350 –400

–300

100 150

321 4 5

f

35. 2.1019

1.6.1019

1.2.1019

8.1018

0

8.1018

2.5.109 7.5.109 x

y

10105.109

37. There is no value for a that will keep the function from having a y-intercept. The absolute value function always crosses the y-intercept when x = 0. 39. ∣ p − 0.08 ∣ ≤ 0.015 41. ∣ x − 5.0 ∣ ≤ 0.01

SeCtion 3.7 1. Each output of a function must have exactly one input for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that y-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no y-values repeat and the function is one-to-one. 3. Yes. For example, f (x) = 1 _ x is its own inverse. 5. y = f −1(x)

7. f −1(x) = x − 3 9. f −1(x) = 2 − x 11. f −1(x) = − 2x _ x − 1 13. Domain of f (x): [−7,∞); f −1 (x) = √

— x − 7

15. Domain of f (x): [0, ∞); f −1 (x) = √ —

x + 5 17. f ( g(x)) = x and g( f (x)) = x 19. One-to-one 21. One-to-one 23. Not one-to-one 25. 3 27. 2

4

x

y

f –1

f

–2–4–6–8–10 –2 –4

2

6

6

42 8

8

10

10

29. 31. [2, 10] 33. 6 35. −4 37. 0 39. 1 41.

x 1 4 7 12 16 f −1(x) 3 6 9 13 14

 

 

ODD ANSWERS C-9

43. f −1 (x) = (1 + x ) 1 _ 3

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

f –1 f

 

45. f −1 (x) = 5 _ 9 (x − 32)

47. t(d) = d _ 50 ; t(180) = 180 _ 50 . The

time for the car to travel 180 miles is 3.6 hours.

Chapter 3 Review exercises 1. Function 3. Not a function 5. f (−3) = −27; f (2) = −2; f (−a) = −2a2 − 3a; −f (a) = 2a2 −3a; f (a + h) = −2a2 − 4ah − 2h2 + 3a + 3h 7. One-to-one 9. Function 11. Function

13.

2

x

y

–1–2–3–4–5 –1 –2 –3

1

3

3

21 4 5

15. 2 17. −1.8 or 1.8

19. −64 + 80a − 16a 2 __

−1 + a

= −16a + 64; a ≠ 1

21. (−∞, −2)∪(−2, 6)∪(6, ∞)

23.

2

x

y

–1–2–3–4–5 –1 –2 –3

1

3

3

21 4 5

25. 31 27. Increasing on (2, ∞), decreasing on (−∞, 2)

29. Increasing on (−3, 1), constant on (−∞, −3) and (1, ∞) 31. Local minimum: (−2, −3); local maximum: (1, 3) 33. Absolute maximum: 10 35. ( f ∘ g )(x) = 17 − 18x, ( g ∘ f )(x) = −7 −18x

37. ( f ∘ g )(x) = √ ______

1 _ x + 2 ; ( g ∘ f )(x) = 1 _

√ —

x + 2

39. ( f ∘ g )(x) = 1 ____ 1 __ x + 1

_

1 ____ 1 __ x + 4

= 1 + x _____

1 + 4x ; Domain:  −∞, − 1 _ 4  ∪  −

1 _ 4 , 0  ∪(0, ∞)

41. ( f ∘ g )(x) = 1 _ √

— x ; Domain: (0, ∞)

43. Many solutions; one possible answer: g(x) = 2x −1 _ 3x + 4 and

f (x) = √ — x .

45.

2

x

y

–1–2–3–4–5–6 –1 –2

1

3

3

21 4

4

5 6

5 6 7 8

47.

2

x

y

–1–2–3–4–5–6 –1 –2

1

3

3

21 4

4

5 6

5 6 7 8

49.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5 51.

x

y

–1–2–3–4 –2 –4 –6 –8

–10 –12 –14 –16 –18 –20 –22 –24

2

321 4 5 6 7 8

53.

2

–1–2–3–4–5–6 –1

1

3

3

21 4

4

5 6

5

x

y

55. f (x) = ∣ x − 3 ∣ 57. Even 59. Odd 61. Even

63. f (x) = 1 _ 2 ∣ x + 2 ∣ + 1 65. f (x) = −3 ∣ x − 3 ∣ + 3

67.

2

x

y

–1–2–3–4 –1 –2 –3 –4 –5

1

3

3 4

21 4 5 76

5

75. 5

69. f −1(x) = x − 9 _ 10

71. f −1(x) = √ —

x − 1

73. The function is one-to-one.

1 2 3 4 5

1 2 3 4 5–1 –2 –3 –4 –5

–1–2–3–4–5 x

y

Chapter 3 practice test 1. Relation is a function 3. −16 5. The graph is a parabola and the graph fails the horizontal line test. 7. 2a2 − a 9. −2(a + b) + 1; b ≠ a 11. √

— 2

13.

1 2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5

3

3

21 4

4

5 6

5

15. Even 17. Odd

19. f −1(x) = x + 5 _ 3

21. (−∞, −1.1) and (1.1, ∞) 23. (1.1, −0.9) 25. f (2) = 2

27. f (x) =  |x| if x ≤ 2 3 if x > 2 29. x = 2 31. Yes

33. f −1(x) = − x − 11 _ 2 or 11 − x _ 2

ChapteR 4

Section 4.1 1. Terry starts at an elevation of 3,000 feet and descends 70 feet per second. 3. d(t) = 100 − 10t 5. The point of intersection is (a, a). This is because for the horizontal line, all of the y-coordinates are a and for the vertical line, all of the x-coordinates are a. The point of intersection is on both lines and therefore will have these two characteristics.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest