Restricting a Domain

In the case in which the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function. Let’s consider the function f(x)=x2−2f(x)=x2−2. It is not one-to-one. The graph is shown at left.

Suppose that we had tried to find a formula for the inverse as follows:

y=x2−2Replacing f(x) with yx=y2−2Interchanging x and yx+2=y2±√x+2=y.Solving for y
yxx+2±x+2−−−−−√====x2−2y2−2y2y.Replacing f(x) with yInterchanging x and ySolving for y

This is not the equation of a function. An input of, say, 2 would yield two outputs, −2 and 2. In such cases, it is convenient to consider “part” of the function by restricting the domain of f(x)f(x). For example, if we restrict the domain of f(x)=x2−2f(x)=x2−2 to nonnegative numbers, then its inverse is a function, as shown at left by the graphs of f(x)=x2−2, x≥0f(x)=x2−2, x≥0, and f−1(x)=√x+2f−1(x)=x+2−−−−−√.

5.1 Exercise Set

Find the inverse of the relation.

  1. 1. {(7, 8), (−2, 8), (3, −4), (8, −8)}{(7, 8), (−2, 8), (3, −4), (8, −8)}

  2. 2. {(0, 1), (5, 6), (−2, −4)}{(0, 1), (5, 6), (−2, −4)}

  3. 3. {(−1, −1), (−3, 4)}{(−1, −1), (−3, 4)}

  4. 4. {(−1, 3), (2, 5), (−3, 5), (2, 0)}{(−1, 3), (2, 5), (−3, 5), (2, 0)}

Find an equation of the inverse relation.

  1. 5. y=4x−5y=4x−5

  2. 6. 2x2+5y2=42x2+5y2=4

  3. 7. x3y=−5x3y=−5

  4. 8. y=3x2−5x+9y=3x2−5x+9

  5. 9. x=y2−2yx=y2−2y

  6. 10. x=12y+4x=12y+4

Graph the equation by substituting and plotting points. Then reflect the graph across the line y=xy=x to obtain the graph of its inverse.

  1. 11. x=y2−3x=y2−3

  2. 12. y=x2+1y=x2+1

  3. 13. y=3x−2y=3x−2

  4. 14. x=−y+4x=−y+4

  5. 15. y=|x|y=|x|

  6. 16. x+2=|y|x+2=|y|

Given the function f, prove that f is one-to-one using the definition of a one-to-one function on p. 314.

  1. 17. f(x)=13x−6f(x)=13x−6

  2. 18. f(x)=4−2xf(x)=4−2x

  3. 19. f(x)=x3+12f(x)=x3+12

  4. 20. f(x)=3√xf(x)=x−−√3

Given the function g, prove that g is not one-to-one using the definition of a one-to-one function on p. 314.

  1. 21. g(x)=1−x2g(x)=1−x2

  2. 22. g(x)=3x2+1g(x)=3x2+1

  3. 23. g(x)=x4−x2g(x)=x4−x2

  4. 24. g(x)=1x6g(x)=1x6

Using the horizontal-line test, determine whether the function is one-to-one.

  1. 25. f(x)=2.7xf(x)=2.7x

  2. 26. f(x)=2−xf(x)=2−x

  3. 27. f(x)=4−x2f(x)=4−x2

  4. 28. f(x)=x3−3x+1f(x)=x3−3x+1

  5. 29. f(x)=8x2−4f(x)=8x2−4

  6. 30. f(x)=√104+xf(x)=104+x−−−√

  7. 31. f(x)=3√x​+​2−2f(x)=x​+​2−−−−−−√3−2

  8. 32. f(x)=8xf(x)=8x

Graph the function and determine whether the function is one-to-one using the horizontal-line test.

  1. 33. f(x)=5x−8f(x)=5x−8

  2. 34. f(x)=3+4xf(x)=3+4x

  3. 35. f(x)=1−x2f(x)=1−x2

  4. 36. f(x)=|x|−2f(x)=|x|−2

  5. 37. f(x)=|x+2|f(x)=|x+2|

  6. 38. f(x)=−0.8f(x)=−0.8

  7. 39. f(x)=−4xf(x)=−4x

  8. 40. f(x)=2x+3f(x)=2x+3

  9. 41. f(x)=23f(x)=23

  10. 42. f(x)=12x2+3f(x)=12x2+3

  11. 43. f(x)=√25−x2f(x)=25−x2−−−−−−√

  12. 44. f(x)=−x3+2f(x)=−x3+2

In Exercises 45–60, for each function:

  1. Determine whether it is one-to-one.

  2. If the function is one-to-one, find a formula for the inverse.

  1. 45. f(x)=x+4f(x)=x+4

  2. 46. f(x)=7−xf(x)=7−x

  3. 47. f(x)=2x−1f(x)=2x−1

  4. 48. f(x)=5x+8f(x)=5x+8

  5. 49. f(x)=4x+7f(x)=4x+7

  6. 50. f(x)=−3xf(x)=−3x

  7. 51. f(x)=x+4x−3f(x)=x+4x−3

  8. 52. f(x)=5x−32x+1f(x)=5x−32x+1

  9. 53. f(x)=x3−1f(x)=x3−1

  10. 54. f(x)=(x+5)3f(x)=(x+5)3

  11. 55. f(x)=x√4−x2f(x)=x4−x2−−−−−√

  12. 56. f(x)=2x2−x−1f(x)=2x2−x−1

  13. 57. f(x)=5x2−2, x≥0f(x)=5x2−2, x≥0

  14. 58. f(x)=4x2+3, x≥0f(x)=4x2+3, x≥0

  15. 59. f(x)=√x+1f(x)=x+1−−−−−√

  16. 60. f(x)=3√x​−​8f(x)=x​−​8−−−−−−√3

Find the inverse by thinking about the operations of the function and then reversing, or undoing, them. Check your work algebraically.

FUNCTION INVERSE
61. f(x)=3xf(x)=3x f−1(x)=f−1(x)=
62. f(x)=14x+7f(x)=14x+7 f−1(x)=f−1(x)=
63. f(x)=−xf(x)=−x f−1(x)=f−1(x)=
64. f(x)=3√x​−​5f(x)=x​−​5−−−−−−√3 f−1(x)=f−1(x)=
65. f(x)=3√x​−​5f(x)=x​−​5−−−−−−√3 f−1(x)=f−1(x)=
66. f(x)=x−1f(x)=x−1 f−1(x)=f−1(x)=

Each graph in Exercises 67–72 is the graph of a one-to-one function f. Sketch the graph of the inverse function f−1f−1.

  1. 67.

  2. 68.

  3. 69.

  4. 70.

  5. 71.

  6. 72.

For the function f, use composition of functions to show that f−1f−1 is as given.

  1. 73. f(x)=78x, f−1(x)=87xf(x)=78x, f−1(x)=87x

  2. 74. f(x)=x+54, f−1(x)=4x−5f(x)=x+54, f−1(x)=4x−5

  3. 75. f(x)=1−xx, f−1(x)=1x+1f(x)=1−xx, f−1(x)=1x+1

  4. 76. f(x)=3√x+4, f−1(x)=x3−4f(x)=x+4−−−−−√3, f−1(x)=x3−4

  5. 77. f(x)=25x+1, f−1(x)=5x−52f(x)=25x+1, f−1(x)=5x−52

  6. 78. f(x)=x+63x−4, f−1(x)=4x+63x−1f(x)=x+63x−4, f−1(x)=4x+63x−1

Find the inverse of the given one-to-one function f. Give the domain and the range of f and of f−1f−1, and then graph both f and f−1f−1 on the same set of axes.

  1. 79. f(x)=5x−3f(x)=5x−3

  2. 80. f(x)=2−xf(x)=2−x

  3. 81. f(x)=2xf(x)=2x

  4. 82. f(x)=−3x+1f(x)=−3x+1

  5. 83. f(x)=13x3−2f(x)=13x3−2

  6. 84. f(x)=3√x−1f(x)=x−−√3−1

  7. 85. f(x)=x+1x−3f(x)=x+1x−3

  8. 86. f(x)=x−1x+2f(x)=x−1x+2

  9. 87. Find f(f−1(5))f(f−1(5)) and f−1(f(a)):f−1(f(a)):

    f(x)=x3−4.
    f(x)=x3−4.

  10. 88. Find (f−1(f(p))(f−1(f(p)) and f(f−1(1253)):f(f−1(1253)):

    f(x)=5√2x​−​73x​+​4.
    f(x)=2x​−​73x​+​4−−−−−−−−√5.

  11. 89. Hitting Lessons. A summer little-league baseball team determines that the cost per player of a group hitting lesson is given by the formula

    C(x)=72+2xx,
    C(x)=72+2xx,

    where x is the number of players in the group and C(x)C(x) is in dollars.

    1. Determine the cost per player of a group hitting lesson when there are 2, 5, and 8 players in the group.

    2. Find a formula for the inverse of the function and explain what it represents.

    3. Use the inverse function to determine the number of players in the group lesson when the cost per player is $74, $20, and $11.

  12. 90. Women’s Shoe Sizes. A function that will convert women’s shoe sizes in the United States to those in Australia is

    s(x)=2x−32
    s(x)=2x−32

    (Source: OnlineConversion.com).

    1. Determine the women’s shoe sizes in Australia that correspond to sizes 5, 7125, 712, and 8 in the United States.

    2. Find a formula for the inverse of the function and explain what it represents.

    3. Use the inverse function to determine the women’s shoe sizes in the United States that correspond to sizes 3, 5123, 512, and 7 in Australia.

  13. 91. E-Commerce Holiday Sales. Retail e-commerce holiday season sales (November and December), in billions of dollars, x years after 2008 is given by the function

    H(x)=6.58x+27.7
    H(x)=6.58x+27.7

    (Source: statista.com).

    1. Determine the total amount of e-commerce holiday sales in 2010 and in 2013.

    2. Find H−1(x)H−1(x) and explain what it represents.

  14. 92. Converting Temperatures. The following formula can be used to convert Fahrenheit temperatures x to Celsius temperatures T(x):T(x):

    T(x)=59(x−32).
    T(x)=59(x−32).

    1. Find T(−13°)T(−13°) and T(86°)T(86°).

    2. Find T−1(x)T−1(x) and explain what it represents.

Skill Maintenance

Consider the quadratic functions (a)–(h) that follow. Without graphing them, answer the questions below. [3.3]

  1. f(x)=2x2f(x)=2x2

  2. f(x)=−x2f(x)=−x2

  3. f(x)=14x2f(x)=14x2

  4. f(x)=−5x2+3f(x)=−5x2+3

  5. f(x)=23(x−1)2−3f(x)=23(x−1)2−3

  6. f(x)=−2(x+3)2+1f(x)=−2(x+3)2+1

  7. f(x)=(x−3)2+1f(x)=(x−3)2+1

  8. f(x)=−4(x+1)2−3f(x)=−4(x+1)2−3

  1. 93. Which functions have a maximum value?

  2. 94. Which graphs open up?

  3. 95. Consider (a) and (c). Which graph is narrower?

  4. 96. Consider (d) and (e). Which graph is narrower?

  5. 97. Which graph has vertex (−3, 1)?

  6. 98. For which is the line of symmetry x=0?x=0?

Synthesis

  1. 99. The function f(x)=x2−3f(x)=x2−3 is not one-to-one. Restrict the domain of f so that its inverse is a function. Find the inverse and state the restriction on the domain of the inverse.

  2. 100. Consider the function f given by

    f(x)={x3+2, for x≤−1,x2, for −1<x<1,x+1, for x≥1.
    f(x)=⎧⎩⎨⎪⎪x3+2,x2,x+1, for x≤−1, for −1<x<1, for x≥1.

    Does f have an inverse that is a function? Why or why not?

  3. 101. Find three examples of functions that are their own inverses; that is, f=f−1f=f−1.

  4. 102. Given the function f(x)=ax+b, a≠0f(x)=ax+b, a≠0, find the values of a and b for which f−1(x)=f(x)f−1(x)=f(x).

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