1.8 Inverse Function Homework from Blitzer #11, 13, 23-27 odds, 29-39, 41-47 odds

240 Chapter 1 Functions and Graphs

In our final example, we will first find Then we will graph and in thesame rectangular coordinate system.

Finding the Inverse of a Domain-Restricted Function

Find the inverse of if Graph and in the same rectangularcoordinate system.

Solution The graph of is the graph of the standard quadraticfunction shifted vertically down 1 unit. Figure 1.71 shows the function’s graph. Thisgraph fails the horizontal line test, so the function does not have aninverse function. By restricting the domain to as given, we obtain a new func-tion whose graph is shown in red in Figure 1.71. This red portion of the graph isincreasing on the interval and passes the horizontal line test.This tells us that

has an inverse function if we restrict its domain to We use ourfour-step procedure to find this inverse function. Begin with Step 1 Replace with Step 2 Interchange and Step 3 Solve for

This is the equation from step 2.

Add 1 to both sides.

Apply the square root property.

Step 4 Replace with

Thus, the inverse of is The graphsof and are shown in Figure 1.72. We obtained the graph of by shifting the graph of the square root function, horizontally to the left1 unit. Note that the green graph of is the reflection of the red graph of aboutthe line

Check Point 7 Find the inverse of if Graph and in the same rectangular coordinate system.

f-1fx Ú 0.f1x2 = x2 + 1

y = x.ff-1

y = 1x ,f-11x2 = 2x + 1f-1f

f-11x2 = 2x + 1.f1x2 = x2 - 1, x Ú 0,

f-11x2 = 2x + 1.f !1(x):y

Because y r 0, take onlythe principal square root andnot the negative square root.

!x+1=y

x + 1 = y2

x = y2 - 1, y Ú 0

y:x = y2 - 1, y Ú 0.y:x

y = x2 - 1, x Ú 0.y:f(x)f1x2 = x2 - 1, x Ú 0.

x Ú 0.f1x2 = x2 - 110, q2 x Ú 0,

f1x2 = x2 - 1

f1x2 = x2 - 1

f-1fx Ú 0.f1x2 = x2 - 1

EXAMPLE 7

f-1ff-1.

x

y

1 2 3 4 5 1

12345

2 3 4 5

1 2 3 4 5

y � x

f(x) = x2 1,x r�0

f 1(x) � !x + 1

Figure 1.72

x

y

1 2 3 4 5 1

12345

2

4 5

1 2 3 4 5

f (x ) � x 2 1

Figure 1.71

! Find the inverse of a functionand graph both functions on thesame axes.

Exercise Set 1.8

Practice ExercisesIn Exercises 1–10, find and and determinewhether each pair of functions and are inverses of each other.

1. and

2. and

3. and

4. and

5. and g1x2 = x + 59

f1x2 = 5x - 9

g1x2 = x - 94

f1x2 = 4x + 9

g1x2 = x - 83

f1x2 = 3x + 8

g1x2 = x6

f1x2 = 6x

g1x2 = x4

f1x2 = 4x

gfg1f1x22f1g1x22 6. and

7. and

8. and

9. and

10. and

The functions in Exercises 11–28 are all one-to-one. For eachfunction,

a. Find an equation for the inverse function.

b. Verify that your equation is correct by showing thatand f-11f1x22 = x.f1f-11x22 = x

f-11x2,g1x2 = x3 + 4f1x2 = 23 x - 4

g1x2 = -xf1x2 = -x

g1x2 = 2x

+ 5f1x2 = 2x - 5

g1x2 = 3x

+ 4f1x2 = 3x - 4

g1x2 = x + 37

f1x2 = 3x - 7

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 240

Section 1.8 Inverse Functions 241

x f(x)

-1 1

0 4

1 5

2 -1

11. 12.13. 14.15. 16.17. 18.19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Which graphs in Exercises 29–34 represent functions that haveinverse functions?

29. 30.

31. 32.

33. 34.

In Exercises 35–38, use the graph of to draw the graph of itsinverse function.

35. 36.

37. 38.

1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

1

1234

2 3 4

1 2 3 54 1 2 3

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

f

y

x

y

x

y

x

y

x

y

x

y

x

f1x2 = 2x - 3x + 1

f1x2 = 2x + 1x - 3

f1x2 = 4x

+ 9f1x2 = 7x

- 3

f1x2 = 13 xf1x2 = 1x

f1x2 = 2x

f1x2 = 1x

f1x2 = 1x - 123f1x2 = 1x + 223 f1x2 = x3 - 1f1x2 = x3 + 2

f1x2 = 3x - 1f1x2 = 2x + 3

f1x2 = 4xf1x2 = 2x

f1x2 = x + 5f1x2 = x + 3

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

x g(x)

-1 0

1 1

4 2

10 -1

53. 54. 55.56. 57. 58.

In Exercises 59–64, let

Evaluate the indicated function without finding an equation forthe function.

59. 60. 61.62. 63. 64.

Application ExercisesThe bar graph shows the average number of hours that Americanssleep per day, by age group. Use this information to solve Exercises 65–66.

f1g3h11242g1f3h11242g-1172 f-11121g ! f21021f ! g2102 h1x2 = x2 + x + 2.

g1x2 = 4x - 1

f1x2 = 2x - 5

f-11g1122f-11g110221g ! f2102 1g ! f21-12f1g1422f1g1122

In Exercises 39–52,

a. Find an equation for

b. Graph and in the same rectangular coordinate system.

c. Use interval notation to give the domain and the range ofand

39. 40.41. 42.43. 44.45. 46.47. 48.

(Hint for Exercises 49–52: To solve for a variable involving an nthroot, raise both sides of the equation to the nth power: )

49. 50.51. 52.

Practice PlusIn Exercises 53–58, and are defined by the following tables. Usethe tables to evaluate each composite function.

gf

f1x2 = 23 x - 1f1x2 = 13 x + 1

f1x2 = 1x + 2f1x2 = 2x - 1

11n y2n = y.

f1x2 = 1x - 223f1x2 = 1x + 223 f1x2 = x3 + 1f1x2 = x3 - 1

f1x2 = 1x - 122, x Ú 1f1x2 = 1x - 122, x … 1

f1x2 = x2 - 1, x … 0f1x2 = x2 - 4, x Ú 0

f1x2 = 2x - 3f1x2 = 2x - 1

f-1.f

f-1f

f-11x2.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 241

Section 1.8 Inverse Functions 241

x f(x)

-1 1

0 4

1 5

2 -1

11. 12.13. 14.15. 16.17. 18.19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Which graphs in Exercises 29–34 represent functions that haveinverse functions?

29. 30.

31. 32.

33. 34.

In Exercises 35–38, use the graph of to draw the graph of itsinverse function.

35. 36.

37. 38.

1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

1

1234

2 3 4

1 2 3 54 1 2 3

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

f

y

x

y

x

y

x

y

x

y

x

y

x

f1x2 = 2x - 3x + 1

f1x2 = 2x + 1x - 3

f1x2 = 4x

+ 9f1x2 = 7x

- 3

f1x2 = 13 xf1x2 = 1x

f1x2 = 2x

f1x2 = 1x

f1x2 = 1x - 123f1x2 = 1x + 223 f1x2 = x3 - 1f1x2 = x3 + 2

f1x2 = 3x - 1f1x2 = 2x + 3

f1x2 = 4xf1x2 = 2x

f1x2 = x + 5f1x2 = x + 3

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

x g(x)

-1 0

1 1

4 2

10 -1

53. 54. 55.56. 57. 58.

In Exercises 59–64, let

Evaluate the indicated function without finding an equation forthe function.

59. 60. 61.62. 63. 64.

Application ExercisesThe bar graph shows the average number of hours that Americanssleep per day, by age group. Use this information to solve Exercises 65–66.

f1g3h11242g1f3h11242g-1172 f-11121g ! f21021f ! g2102 h1x2 = x2 + x + 2.

g1x2 = 4x - 1

f1x2 = 2x - 5

f-11g1122f-11g110221g ! f2102 1g ! f21-12f1g1422f1g1122

In Exercises 39–52,

a. Find an equation for

b. Graph and in the same rectangular coordinate system.

c. Use interval notation to give the domain and the range ofand

39. 40.41. 42.43. 44.45. 46.47. 48.

(Hint for Exercises 49–52: To solve for a variable involving an nthroot, raise both sides of the equation to the nth power: )

49. 50.51. 52.

Practice PlusIn Exercises 53–58, and are defined by the following tables. Usethe tables to evaluate each composite function.

gf

f1x2 = 23 x - 1f1x2 = 13 x + 1

f1x2 = 1x + 2f1x2 = 2x - 1

11n y2n = y.

f1x2 = 1x - 223f1x2 = 1x + 223 f1x2 = x3 + 1f1x2 = x3 - 1

f1x2 = 1x - 122, x Ú 1f1x2 = 1x - 122, x … 1

f1x2 = x2 - 1, x … 0f1x2 = x2 - 4, x Ú 0

f1x2 = 2x - 3f1x2 = 2x - 1

f-1.f

f-1f

f-11x2.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 241

Section 1.8 Inverse Functions 241

x f(x)

-1 1

0 4

1 5

2 -1

11. 12.13. 14.15. 16.17. 18.19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Which graphs in Exercises 29–34 represent functions that haveinverse functions?

29. 30.

31. 32.

33. 34.

In Exercises 35–38, use the graph of to draw the graph of itsinverse function.

35. 36.

37. 38.

1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

1

1234

2 3 4

1 2 3 54 1 2 3

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

f

y

x

y

x

y

x

y

x

y

x

y

x

f1x2 = 2x - 3x + 1

f1x2 = 2x + 1x - 3

f1x2 = 4x

+ 9f1x2 = 7x

- 3

f1x2 = 13 xf1x2 = 1x

f1x2 = 2x

f1x2 = 1x

f1x2 = 1x - 123f1x2 = 1x + 223 f1x2 = x3 - 1f1x2 = x3 + 2

f1x2 = 3x - 1f1x2 = 2x + 3

f1x2 = 4xf1x2 = 2x

f1x2 = x + 5f1x2 = x + 3

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

x g(x)

-1 0

1 1

4 2

10 -1

53. 54. 55.56. 57. 58.

In Exercises 59–64, let

Evaluate the indicated function without finding an equation forthe function.

59. 60. 61.62. 63. 64.

Application ExercisesThe bar graph shows the average number of hours that Americanssleep per day, by age group. Use this information to solve Exercises 65–66.

f1g3h11242g1f3h11242g-1172 f-11121g ! f21021f ! g2102 h1x2 = x2 + x + 2.

g1x2 = 4x - 1

f1x2 = 2x - 5

f-11g1122f-11g110221g ! f2102 1g ! f21-12f1g1422f1g1122

In Exercises 39–52,

a. Find an equation for

b. Graph and in the same rectangular coordinate system.

c. Use interval notation to give the domain and the range ofand

39. 40.41. 42.43. 44.45. 46.47. 48.

(Hint for Exercises 49–52: To solve for a variable involving an nthroot, raise both sides of the equation to the nth power: )

49. 50.51. 52.

Practice PlusIn Exercises 53–58, and are defined by the following tables. Usethe tables to evaluate each composite function.

gf

f1x2 = 23 x - 1f1x2 = 13 x + 1

f1x2 = 1x + 2f1x2 = 2x - 1

11n y2n = y.

f1x2 = 1x - 223f1x2 = 1x + 223 f1x2 = x3 + 1f1x2 = x3 - 1

f1x2 = 1x - 122, x Ú 1f1x2 = 1x - 122, x … 1

f1x2 = x2 - 1, x … 0f1x2 = x2 - 4, x Ú 0

f1x2 = 2x - 3f1x2 = 2x - 1

f-1.f

f-1f

f-11x2.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 241

Section 1.8 Inverse Functions 241

x f(x)

-1 1

0 4

1 5

2 -1

11. 12.13. 14.15. 16.17. 18.19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Which graphs in Exercises 29–34 represent functions that haveinverse functions?

29. 30.

31. 32.

33. 34.

In Exercises 35–38, use the graph of to draw the graph of itsinverse function.

35. 36.

37. 38.

1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

1

1234

2 3 4

1 2 3 54 1 2 3

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

f

y

x

y

x

y

x

y

x

y

x

y

x

f1x2 = 2x - 3x + 1

f1x2 = 2x + 1x - 3

f1x2 = 4x

+ 9f1x2 = 7x

- 3

f1x2 = 13 xf1x2 = 1x

f1x2 = 2x

f1x2 = 1x

f1x2 = 1x - 123f1x2 = 1x + 223 f1x2 = x3 - 1f1x2 = x3 + 2

f1x2 = 3x - 1f1x2 = 2x + 3

f1x2 = 4xf1x2 = 2x

f1x2 = x + 5f1x2 = x + 3

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

x g(x)

-1 0

1 1

4 2

10 -1

53. 54. 55.56. 57. 58.

In Exercises 59–64, let

Evaluate the indicated function without finding an equation forthe function.

59. 60. 61.62. 63. 64.

Application ExercisesThe bar graph shows the average number of hours that Americanssleep per day, by age group. Use this information to solve Exercises 65–66.

f1g3h11242g1f3h11242g-1172 f-11121g ! f21021f ! g2102 h1x2 = x2 + x + 2.

g1x2 = 4x - 1

f1x2 = 2x - 5

f-11g1122f-11g110221g ! f2102 1g ! f21-12f1g1422f1g1122

In Exercises 39–52,

a. Find an equation for

b. Graph and in the same rectangular coordinate system.

c. Use interval notation to give the domain and the range ofand

39. 40.41. 42.43. 44.45. 46.47. 48.

(Hint for Exercises 49–52: To solve for a variable involving an nthroot, raise both sides of the equation to the nth power: )

49. 50.51. 52.

Practice PlusIn Exercises 53–58, and are defined by the following tables. Usethe tables to evaluate each composite function.

gf

f1x2 = 23 x - 1f1x2 = 13 x + 1

f1x2 = 1x + 2f1x2 = 2x - 1

11n y2n = y.

f1x2 = 1x - 223f1x2 = 1x + 223 f1x2 = x3 + 1f1x2 = x3 - 1

f1x2 = 1x - 122, x Ú 1f1x2 = 1x - 122, x … 1

f1x2 = x2 - 1, x … 0f1x2 = x2 - 4, x Ú 0

f1x2 = 2x - 3f1x2 = 2x - 1

f-1.f

f-1f

f-11x2.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 241

Section 1.8 Inverse Functions 241

x f(x)

-1 1

0 4

1 5

2 -1

11. 12.13. 14.15. 16.17. 18.19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Which graphs in Exercises 29–34 represent functions that haveinverse functions?

29. 30.

31. 32.

33. 34.

In Exercises 35–38, use the graph of to draw the graph of itsinverse function.

35. 36.

37. 38.

1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

1

1234

2 3 4

1 2 3 54 1 2 3

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

f

y

x

y

x

y

x

y

x

y

x

y

x

f1x2 = 2x - 3x + 1

f1x2 = 2x + 1x - 3

f1x2 = 4x

+ 9f1x2 = 7x

- 3

f1x2 = 13 xf1x2 = 1x

f1x2 = 2x

f1x2 = 1x

f1x2 = 1x - 123f1x2 = 1x + 223 f1x2 = x3 - 1f1x2 = x3 + 2

f1x2 = 3x - 1f1x2 = 2x + 3

f1x2 = 4xf1x2 = 2x

f1x2 = x + 5f1x2 = x + 3

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

x g(x)

-1 0

1 1

4 2

10 -1

53. 54. 55.56. 57. 58.

In Exercises 59–64, let

Evaluate the indicated function without finding an equation forthe function.

59. 60. 61.62. 63. 64.

Application ExercisesThe bar graph shows the average number of hours that Americanssleep per day, by age group. Use this information to solve Exercises 65–66.

f1g3h11242g1f3h11242g-1172 f-11121g ! f21021f ! g2102 h1x2 = x2 + x + 2.

g1x2 = 4x - 1

f1x2 = 2x - 5

f-11g1122f-11g110221g ! f2102 1g ! f21-12f1g1422f1g1122

In Exercises 39–52,

a. Find an equation for

b. Graph and in the same rectangular coordinate system.

c. Use interval notation to give the domain and the range ofand

39. 40.41. 42.43. 44.45. 46.47. 48.

(Hint for Exercises 49–52: To solve for a variable involving an nthroot, raise both sides of the equation to the nth power: )

49. 50.51. 52.

Practice PlusIn Exercises 53–58, and are defined by the following tables. Usethe tables to evaluate each composite function.

gf

f1x2 = 23 x - 1f1x2 = 13 x + 1

f1x2 = 1x + 2f1x2 = 2x - 1

11n y2n = y.

f1x2 = 1x - 223f1x2 = 1x + 223 f1x2 = x3 + 1f1x2 = x3 - 1

f1x2 = 1x - 122, x Ú 1f1x2 = 1x - 122, x … 1

f1x2 = x2 - 1, x … 0f1x2 = x2 - 4, x Ú 0

f1x2 = 2x - 3f1x2 = 2x - 1

f-1.f

f-1f

f-11x2.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 241

Section 1.8 Inverse Functions 241

x f(x)

-1 1

0 4

1 5

2 -1

11. 12.13. 14.15. 16.17. 18.19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Which graphs in Exercises 29–34 represent functions that haveinverse functions?

29. 30.

31. 32.

33. 34.

In Exercises 35–38, use the graph of to draw the graph of itsinverse function.

35. 36.

37. 38.

1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

1

1234

2 3 4

1 2 3 54 1 2 3

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

f

y

x

y

x

y

x

y

x

y

x

y

x

f1x2 = 2x - 3x + 1

f1x2 = 2x + 1x - 3

f1x2 = 4x

+ 9f1x2 = 7x

- 3

f1x2 = 13 xf1x2 = 1x

f1x2 = 2x

f1x2 = 1x

f1x2 = 1x - 123f1x2 = 1x + 223 f1x2 = x3 - 1f1x2 = x3 + 2

f1x2 = 3x - 1f1x2 = 2x + 3

f1x2 = 4xf1x2 = 2x

f1x2 = x + 5f1x2 = x + 3

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

x g(x)

-1 0

1 1

4 2

10 -1

53. 54. 55.56. 57. 58.

In Exercises 59–64, let

Evaluate the indicated function without finding an equation forthe function.

59. 60. 61.62. 63. 64.

Application ExercisesThe bar graph shows the average number of hours that Americanssleep per day, by age group. Use this information to solve Exercises 65–66.

f1g3h11242g1f3h11242g-1172 f-11121g ! f21021f ! g2102 h1x2 = x2 + x + 2.

g1x2 = 4x - 1

f1x2 = 2x - 5

f-11g1122f-11g110221g ! f2102 1g ! f21-12f1g1422f1g1122

In Exercises 39–52,

a. Find an equation for

b. Graph and in the same rectangular coordinate system.

c. Use interval notation to give the domain and the range ofand

39. 40.41. 42.43. 44.45. 46.47. 48.

(Hint for Exercises 49–52: To solve for a variable involving an nthroot, raise both sides of the equation to the nth power: )

49. 50.51. 52.

Practice PlusIn Exercises 53–58, and are defined by the following tables. Usethe tables to evaluate each composite function.

gf

f1x2 = 23 x - 1f1x2 = 13 x + 1

f1x2 = 1x + 2f1x2 = 2x - 1

11n y2n = y.

f1x2 = 1x - 223f1x2 = 1x + 223 f1x2 = x3 + 1f1x2 = x3 - 1

f1x2 = 1x - 122, x Ú 1f1x2 = 1x - 122, x … 1

f1x2 = x2 - 1, x … 0f1x2 = x2 - 4, x Ú 0

f1x2 = 2x - 3f1x2 = 2x - 1

f-1.f

f-1f

f-11x2.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 241

Section 1.8 Inverse Functions 241

x f(x)

-1 1

0 4

1 5

2 -1

11. 12.13. 14.15. 16.17. 18.19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Which graphs in Exercises 29–34 represent functions that haveinverse functions?

29. 30.

31. 32.

33. 34.

In Exercises 35–38, use the graph of to draw the graph of itsinverse function.

35. 36.

37. 38.

1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

1

1234

2 3 4

1 2 3 54 1 2 3

y

x 1

1234

2 3 4

1 2 3 4 1 2 3 4

y

x

f

y

x

y

x

y

x

y

x

y

x

y

x

f1x2 = 2x - 3x + 1

f1x2 = 2x + 1x - 3

f1x2 = 4x

+ 9f1x2 = 7x

- 3

f1x2 = 13 xf1x2 = 1x

f1x2 = 2x

f1x2 = 1x

f1x2 = 1x - 123f1x2 = 1x + 223 f1x2 = x3 - 1f1x2 = x3 + 2

f1x2 = 3x - 1f1x2 = 2x + 3

f1x2 = 4xf1x2 = 2x

f1x2 = x + 5f1x2 = x + 3

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

x g(x)

-1 0

1 1

4 2

10 -1

53. 54. 55.56. 57. 58.

In Exercises 59–64, let

Evaluate the indicated function without finding an equation forthe function.

59. 60. 61.62. 63. 64.

Application ExercisesThe bar graph shows the average number of hours that Americanssleep per day, by age group. Use this information to solve Exercises 65–66.

f1g3h11242g1f3h11242g-1172 f-11121g ! f21021f ! g2102 h1x2 = x2 + x + 2.

g1x2 = 4x - 1

f1x2 = 2x - 5

f-11g1122f-11g110221g ! f2102 1g ! f21-12f1g1422f1g1122

In Exercises 39–52,

a. Find an equation for

b. Graph and in the same rectangular coordinate system.

c. Use interval notation to give the domain and the range ofand

39. 40.41. 42.43. 44.45. 46.47. 48.

(Hint for Exercises 49–52: To solve for a variable involving an nthroot, raise both sides of the equation to the nth power: )

49. 50.51. 52.

Practice PlusIn Exercises 53–58, and are defined by the following tables. Usethe tables to evaluate each composite function.

gf

f1x2 = 23 x - 1f1x2 = 13 x + 1

f1x2 = 1x + 2f1x2 = 2x - 1

11n y2n = y.

f1x2 = 1x - 223f1x2 = 1x + 223 f1x2 = x3 + 1f1x2 = x3 - 1

f1x2 = 1x - 122, x Ú 1f1x2 = 1x - 122, x … 1

f1x2 = x2 - 1, x … 0f1x2 = x2 - 4, x Ú 0

f1x2 = 2x - 3f1x2 = 2x - 1

f-1.f

f-1f

f-11x2.

P-BLTZMC01_135-276-hr 17-11-2008 11:46 Page 241