Find an equation for the inverse of the relation y = 2x º 4.

STUDENTHELP                                                           SOLUTION

_{Look Back}                                                         y = 2x º 4           Write original relation.

 The inverse relation is y = x + 2.

2

. . . . . . . . . .

In Example 1 both the original relation and the inverse relation happen to be functions. In such cases the two functions are called  inverse functions.

Verify that ƒ(x) = 2x º 4 and ƒ^º1(x) = ¹x + 2 are inverses.

2

SOLUTION Show that ƒ(ƒ^º1(x)) = x and ƒ^º1(ƒ(x)) = x.

                                                                                ƒ(ƒ^º1(x)) = ƒ¹x + 2            ƒ^º1(ƒ(x)) = ƒ^º1(2x º 4)

2

                                                                                                        1                                           1

                                                                                                   = 2x + 2 º 4                              = (2x º 4) + 2

                                                                                                        2                                           2

Writing an Inverse Model

STUDENTHELP

Study Tip

Notice that you do not switch the variables when you are finding inverses for models. This would be confusing because the letters are chosen to remind you of the real-life quantities they represent.

the spring stretches based on given weights. Hooke’s law states that the length a spring stretches is proportional to the weight attached to the spring. A model for one scale is ¬ = 0.5w + 3 where ¬ is the total length (in inches) of the spring and w is the weight (in pounds) of the object.

a.   Find the inverse model for the scale.

b.   If you place a melon on the scale and the spring stretches to a total length of 5.5 inches, how much does the melon weigh?

SOLUTION

a.                     ¬ = 0.5w + 3   Write original model.

             ¬ º 3 = 0.5w                Subtract 3 from each side.

¬ º 3

                   = w                        Divide each side by 0.5.

0.5

          2¬ º 6 = w                    Simplify.

b.                     To find the weight of the melon, substitute 5.5 for ¬. w = 2¬ º 6 = 2(5.5) º 6 = 11 º 6 = 5  The melon weighs 5 pounds.

STUDENTHELP

Look Back 

For help with recognizing when a relationship is a function, see p. 70.

with their reflections in the line y = x. Notice that the inverse of g(x) = x³ is a function, but that the inverse of ƒ(x) = x² is not a function.

If the domain of ƒ(x) = x² is restricted, say to only nonnegative real numbers, then the inverse of ƒ is a function.

EXAMPLE 4              Finding an Inverse Power Function

Find the inverse of the function ƒ(x) = x², x ≥ 0.

SOLUTION

Because the domain of ƒ is restricted to nonnegative values, the inverse function is ƒ^º1(x) = x . (You would choose ƒ^º1(x) = ºx if the domain had been restricted to x ≤ 0.)

✓CHECK To check your work, graph ƒ and ƒ^º1 as shown.

Note that the graph of ƒ^º1(x) = x is the reflection of the graph of ƒ(x) = x², x ≥ 0 in the line y = x.

. . . . . . . . . .

In the graphs at the top of the page, notice that the graph of ƒ(x) = x² can be intersected twice with a horizontal line and that its inverse is not a function. On the other hand, the graph of g(x) = x³ cannot be intersected twice with a horizontal line and its inverse is a function. This observation suggests the horizontal line test.

ASTRONOMY 

The Ring Nebula is part of the constellation Lyra. The radius of the nebula is expanding at an average rate of about 5.99 10⁸ kilometers per year.

              APPLICATION LINK www.mcdougallittell.com

1 ³

Consider the function ƒ(x) = x º 2. Determine whether the inverse of ƒ is a 2

function. Then find the inverse.

SOLUTION

Begin by graphing the function and noticing that no horizontal line intersects the graph more than once. This tells you that the inverse of ƒ is itself a function. To find an equation for ƒ^º1, complete the following steps.

1 ³

                       ƒ(x) = x º 2            Write original function.

2

1 ³

                            y = x º 2            Replace ƒ(x) with y.

ASTRONOMY Near the end of a star’s life the star will eject gas, forming a planetary nebula. The Ring Nebula is an example of a planetary nebula. The volume V (in cubic kilometers) of this nebula can be modeled by V = (9.01 ª 10²⁶)t³ where t is the age (in years) of the nebula. Write the inverse model that gives the age of the nebula as a function of its volume. Then determine the approximate age of the Ring Nebula given that its volume is about 1.5 ª 10³⁸ cubic kilometers.

SOLUTION

                            3 V            = t                              Take cube root of each side.

9.01 ª 10²⁶

             (1.04 ª 10^º9)³V = t                                Simplify.

To find the age of the nebula, substitute 1.5 ª 10³⁸ for V.

            t = (1.04 ª 10^º9)³ V                                Write inverse model.

                                Substitute for V.

           ≈ 5500                                                 Use a calculator.

 The Ring Nebula is about 5500 years old.

GUIDED PRACTICE

4.

Find an equation for the inverse relation.

2

                                                            6. y = 5x                               7. y = 2x º 1                            8. y = ºx + 6

3 Verify that ƒ and g are inverse functions.

                                                                                    ³ g(x) = x1/3                                      10. ƒ(x) = 6x + 3, g(x) = 1x º 1

9. ƒ(x) = 8x ,

                                                                                                       2                                                                6       2

Find the inverse function.

                                                           11. ƒ(x) = 3x⁴, x ≥ 0                                   12. ƒ(x) = 2x³ + 1

13. The graph of ƒ(x) = º|x| + 1 is shown. Is the inverse of ƒ a function? Explain.

Ex. 13

PRACTICE _ANDAPPLICATIONS

Extra Practice

to help you master         14. skills is on p. 949.

FINDING INVERSES Find an equation for the inverse relation.

1

                                                            16. y = º2x + 5                      17. y = 3x º 3                         18. y = x + 6

2

4

                                                             19. y = ºx + 11                    20. y = 11x º 5                       21. y = º12x + 7

5

                                                                                    1                                                                             3       5

                                                             22. y = 3x º                                 23. y = 8x º 13                          24. y = ºx +

                                                                                    4                                                                             7       7

VERIFYING INVERSES Verify that ƒ and g are inverse functions. STUDENTHELP

                                                                                                                                                                                              1         1

       HOMEWORK HELP                          25. ƒ(x) = x + 7, g(x) = x º 7                        26. ƒ(x) = 3x º 1, g(x) = 3x + 3

Example 1: Exs. 14–24

Example 2: Exs. 25–32                      27. ƒ(x) = 1x + 1, g(x) = 2x º 2                          28. ƒ(x) = º2x + 4, g(x) = º1x + 2

Example 3: Exs. 57–59                                                                                                  22

Example 4: Exs. 33–41 ³ g(x) = x11/3 Example 5: Exs. 42–56 29. ƒ(x) = 3x + 1, 3

Example 6: Exs. 60–62                                          5                                                                                                                             4^x

x 7+ 2     ⁵ 7x º2   32. ƒ(x) = 256x⁴, x ≥ 0; g(x) = ₄ 31. ƒ(x) = , g(x) =

Investment bankers have a wide variety of job descriptions. Some buy and sell international currencies at reported exchange rates, discussed in Ex. 57.

N

             CAREER LINK www.mcdougallittell.com VISUAL THINKING Match the graph with the graph of its inverse.

33.34.35.

A.B.C.

INVERSES OF POWER FUNCTIONS Find the inverse power function.

36. ƒ(x) = x⁷                                                         37. ƒ(x) = ºx⁶, x ≥ 0              38. ƒ(x) = 3x⁴, x ≤ 0

39. ƒ(x) = 1x⁵                                                  40. ƒ(x) = 10x³                                               41. ƒ(x) = º^{9 2}

                      32                                                                                                               4

INVERSES OF NONLINEAR FUNCTIONS Find the inverse function.

42. ƒ(x) = x³ + 2                      43. ƒ(x) = º2                  44. ƒ(x) = 2 º 2x², x ≤ 0

45. ƒ(x) = 3x³ º 9                     46. ƒ(x) = x^{4   1} ≥ 0         47. ƒ(x) = ¹x⁵ + ²

                      5                                                                2                                           6           3

HORIZONTAL LINE TEST Graph the function ƒ. Then use the graph to determine whether the inverse of ƒ is a function.

57.  EXCHANGE RATE The Federal Reserve Bank of New York reports

international exchange rates at 12:00 noon each day. On January 20, 1999, the exchange rate for Canada was 1.5226. Therefore, the formula that gives Canadian dollars in terms of United States dollars on that day is

D_C = 1.5226D_US

where D_C represents Canadian dollars and D_US represents United States dollars. Find the inverse of the function to determine the value of a United States dollar in terms of Canadian dollars on January 20, 1999.

DATA UPDATE of Federal Reserve Bank of New York data at www.mcdougallittell.com

58.TEMPERATURE CONVERSION The formula to convert temperatures from degrees Fahrenheit to degrees Celsius is:

5

C = (F º 32) 9

Write the inverse of the function, which converts temperatures from degrees Celsius to degrees Fahrenheit. Then find the Fahrenheit temperatures that are equal to 29°C, 10°C, and 0°C.

STUDENTHELP

      ^N        HOMEWORK HELP

Visit our Web site www.mcdougallittell.com for help with problem solving in Ex. 62.

Test

Preparation

★Challenge

EXTRACHALLENGE

www.mcdougallittell.com

59.    BOWLING In bowling a handicap is a change in score to adjust for differences in players’ abilities. You belong to a bowling league in which each bowler’s handicap h is determined by his or her average a using this formula:

h = 0.9(200 º a)

(If the bowler’s average is over 200, the handicap is 0.) Find the inverse of the function. Then find your average if your handicap is 27.

60.    GAMES You and a friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add 3. If your friend’s final answer is 53, what was the original number chosen? Use an inverse function in your solution.

61.    FISH The weight w (in kilograms) of a hake, a

type of fish, is related to its length l (in centimeters) by this function:

w = (9.37 ª 10^º6)l³

Find the inverse of the function. Then determine the approximate length of a hake that weighs

         0.679 kilogram.   Source: Fishbyte                                             Hake

62.    SHELVES The weight w (in pounds) that can be supported by a shelf made from half-inch Douglas fir plywood can be modeled by

82.9 ³ d

w =

where d is the distance (in inches) between the supports for the shelf. Find the inverse of the function. Then find the distance between the supports of a shelf that can hold a set of encyclopedias weighing 66 pounds.

QUANTITATIVE COMPARISON In Exercises 63 and 64, choose the statement that is true about the given quantities.

¡A The quantity in column A is greater.

          ¡B   The quantity in column B is greater.

          ¡C   The two quantities are equal.

         ¡D   The relationship cannot be determined from the given information.

63.

64.

INVERSE FUNCTIONS Complete Exercises 65–68 to explore functions that are their own inverses.

65.   VISUAL THINKING The functions ƒ(x) = x and g(x) = ºx are their own inverses. Graph each function and explain why this is true.

66.   Graph other linear functions that are their own inverses.

67.   Write equations of the lines you graphed in Exercise 66.

68.   Use your equations from Exercise 67 to find a general formula for a family of linear equations that are their own inverses.

MIXED REVIEW

ABSOLUTE VALUE FUNCTIONS Graph the absolute value function. 

(Review 2.8 for 7.5)

                                                             69. ƒ(x) = |x| º 1                                        70. ƒ(x) = 2|x| + 7

                                                             71. ƒ(x) = |x º 4| + 5                                    72. ƒ(x) = º3|x + 2| º 7

QUADRATIC FUNCTIONS Graph the quadratic function.  (Review 5.1 for 7.5)

                                                           73. ƒ(x) = x² + 2                                         74. ƒ(x) = (x + 3)² º 7

                                                            75. ƒ(x) = 2(x + 2)² º 5                                76. ƒ(x) = º3(x º 4)² + 1

SIMPLIFYING EXPRESSIONS Simplify the expression. Assume all variables are positive.  (Review 7.2)

                                                           77. 4 20 • 4 4⁵                                  78. 191/6 191/3                     79. (5(5yy))61//55

                                                           80. 6 2x⁶                                           81. 3⁷5 + 2⁷5                               82. ³ 270 + 2³ 10

83.  SNACK FOODS Delia, Ruth, and Amy go to the store to buy snacks. Delia buys 3 bagels and 3 apples. Ruth buys 1 pretzel, 2 bagels, and 3 apples. Amy buys 2 pretzels and 4 bagels. Delia’s bill comes to $3.72, Ruth’s to $5.06, and

Amy’s to $6.58. How much does one bagel cost?  (Review 3.6)

QUIZ 2                                                                       Self-Test for Lessons 7.3 and 7.4

Let ƒ(x) = 6x² º x^1/2 and g(x) = 2x^1/2. Perform the indicated operation and state the domain.  (Lesson 7.3)

ƒ(x)

                                                            1. ƒ(x) + g(x)             2. ƒ(x) º g(x)             3. ƒ(x) • g(x)                 4.

g(x)

Let ƒ(x) = 3x^º1 and g(x) = x º 8. Perform the indicated operation and state the domain.  (Lesson 7.3)

                                                           5. ƒ(g(x))                  6. g(ƒ(x))                  7. ƒ(ƒ(x))                  8. g(g(x))

Verify that ƒ and g are inverse functions.  (Lesson 7.4)

                                                                                                           1         3                                                         ^1/3 g(x) = x³ º 1

                                                                  9. ƒ(x) = 2x º 3, g(x) = x +                           10. ƒ(x) = (x + 1)     ,

                                                                                                           2         2

Find the inverse function.  (Lesson 7.4)

                                                          11. ƒ(x) = x + 8                     12. ƒ(x) = 2x⁴, x ≤ 0                13. ƒ(x) = ºx⁵ + 6

Graph the function ƒ. Then use the graph to determine whether the inverse of ƒ is a function.  (Lesson 7.4)

                                                           14. ƒ(x) = 3x⁶ + 2                    15. ƒ(x) = º2x⁵ + 3x º 1           16. ƒ(x) = 6³ x +4

17.  RIPPLES IN A POND You drop a pebble into a calm pond causing ripples of concentric circles. The radius r (in feet) of the outer ripple is given by r(t) = 0.6t where t is the time (in seconds) after the pebble hits the water. The area A (in square feet) of the outer ripple is given by A(r) = πr². Use composition of functions to find the relationship between area and time. Then find the area of the outer ripple after 2 seconds.  (Lesson 7.3)