Кері функция ұғымы. Алгебра, 10 сынып, қосымша материал.

7. 4

What you should learn

nonlinear functions, as applied in Example 6.

Why you should learn it

 To solve real-life problems, such as finding your bowling average

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Inverse Functions

GOAL 1FINDING INVERSES OF LINEAR FUNCTIONS

In Lesson 2.1 you learned that a relation is a mapping of input values onto output values. An inverse relation maps the output values back to their original input values. This means that the domain of the inverse relation is the range of the original relation and that the range of the inverse relation is the domain of the original relation.

The graph of an inverse relation is the reflection of the graph of the original relation.

The line of reflection is y = x.

Graph of original relation

Reflection in y = x

y

x

Graph of inverse relation

To find the inverse of a relation that is given by an equation in x and y, switch the roles of x and y and solve for y (if possible).

E XA M PLE1 Finding an Inverse Relation

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

STUDENT HELP

Look Back

For help with solving equations for y, see p. 26.

SOLUTION

y = 2x º 4Write original relation.

x = 2y º 4Switch x and y.

x + 4 = 2yAdd 4 to each side.

¹ x + 2 = yDivide each side by 2.

2

¹

The inverse relation is y =x + 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.

• Chapter 7 Powers, Roots, and Radicals

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STUDENT HELP

Study Tip

The notation for an inverse function, ƒ^º1, looks like a negative exponent, but it should not be interpreted that way. In other words,

Science

INVERSE FUNCTIONS

Functions ƒ and g are inverses of each other provided:

ƒ(g(x)) = xandg(ƒ(x)) = x

The function g is denoted by ƒ^º1, read as “ƒ inverse.”

Given any function, you can always find its inverse relation by switching x and y. For a linear function ƒ(x) = mx + b where m ≠ 0, the inverse is itself a linear function.

E XA M PLE2 Verifying Inverse Functions

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

2

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

ƒ(ƒ^º1(x)) = ƒ ¹ x + 2ƒ^º1(ƒ(x)) = ƒ^º1(2x º 4)

2

= 2 ¹ x + 2 º 4= ¹ (2x º 4) + 2

22

= x + 4 º 4= x º 2 + 2

= x ✓= x ✓

E XA M PLE3 Writing an Inverse Model

STUDENT HELP

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

SOLUTION

• ¬ = 0.5w + 3

• • º 3 = 0.5w

• º 3

 = w 0.5

2¬ º 6 = w

Write original model.

Subtract 3 from each side.

Divide each side by 0.5.

Simplify.

chosen to remind you of the real-life quantities they represent.

• 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.

7.4 Inverse Functions423

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GOAL 2FINDING INVERSES OF NONLINEAR FUNCTIONS

The graphs of the power functions ƒ(x) = x² and g(x) = x³ are shown below along 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.

STUDENT HELP

Look Back

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

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

E XA M P L E4 Finding an Inverse Power Function

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

x = yTake square roots of each side.

Because the domain of ƒ is restricted to nonnegative values, the inverse function is ƒ^º1(x) = x . (You would choose ƒ^º1(x) = º x 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.

y

ƒ (x)x ²

x ≥ 0

1x

..........

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.

HORIZONTAL LINE TEST

If no horizontal line intersects the graph of a function ƒ more than once, then the inverse of ƒ is itself a function.

• Chapter 7 Powers, Roots, and Radicals

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E XA M P L E5 Finding an Inverse Function

Consider the function ƒ(x) = ¹ x³ 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.

The inverse function is ƒ^º1(x) = ³ 2x + 4 .

E XA M P L E6 Writing an Inverse Model

FOCUS ON

APPLICATIONS

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)t ³ 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

V = (9.01 ª 10²⁶)t³

V

= t₃

9.01 ª 10²⁶

^V= t

3

9.01 ª 10²⁶

(1.04 ª 10^º9) ³ V = t

Write original model.

Isolate power.

Take cube root of each side.

Simplify.

 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.

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

The Ring Nebula is about 5500 years old.

7.4 Inverse Functions425

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GUIDED PRACTICE

Vocabulary Check ✓1. Explain how to use the horizontal line test to determine if an inverse relation is an inverse function.

Concept Check ✓2. Describe how the graph of a relation and the graph of its inverse are related.

• Explain the steps in finding an equation for an inverse function.

Find the inverse function.

11. ƒ(x) = 3x⁴, x ≥ 012. ƒ(x) = 2x³ + 1

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

y

3 x

1

Ex. 13

PRACTICE AND APPLICATIONS

FINDING INVERSES Find an equation for the inverse relation.

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 14–24

Example 2: Exs. 25–32

Example 3: Exs. 57–59

Example 4: Exs. 33–41

Example 5: Exs. 42–56

Example 6: Exs. 60–62

• Chapter 7 Powers, Roots, and Radicals

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x

FOCUS ON

CAREERS

BANKER

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

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

• 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.

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

• = ⁵₉(F º 32)

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.

7.4 Inverse Functions427

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STUDENT HELP

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

Test

Preparation

★ Challenge

EXTRA CHALLENGE

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• 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.

• 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.

• 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

• 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

w =_d

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.

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

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

• Graph other linear functions that are their own inverses.

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

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

• Chapter 7 Powers, Roots, and Radicals

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MIXED REVIEW

ABSOLUTE VALUE FUNCTIONS Graph the absolute value function.

(Review 2.8 for 7.5)

QUIZ 2

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

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

• 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)

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)

9. ƒ(x) = 2x º 3, g(x) = ¹₂x + ³₂10. ƒ(x) = (x + 1)^1/3, g(x) = x³ º 1

Find the inverse function. (Lesson 7.4)

11. ƒ(x) = x + 812. ƒ(x) = 2x⁴, x ≤ 013. 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 + 215. ƒ(x) = º2x⁵ + 3x º 116. ƒ(x) = 6 ³ x + 4

• 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)

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