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5•2 Inverse Functions and Relations

Find Inverses

You have studied inverse operations such as multiplication and division. The inverse of a relation or function can be found algebraically. The graphs of inverse functions are reflections about the line y = x.

Inverse RelationsTwo relations are inverse relations if and only if whenever

one relation contains the element (a, b), the other

relation contains the element (b, a).

Inverse Functions

Two functions f and g are inverse functions if and only if

both [f ◦ g](x) and [g ◦ f ](x) are the identity function.

Find and Graph an Inverse

Find the inverse of the function f (x) = 2 _ 5 x - 1 _ 5 . Then graph

the function and its inverse.

Step 1: Replace f(x) with y in the original equation.

f(x) = 2 _ 5 x - 1 _ 5

y = 2 _ 5 x - 1 _ 5

Step 2: Interchange x and y.

x = 2 _ 5 y - 1 _ 5

Step 3: Solve for y.

x = 2 _ 5 y - 1 _ 5 5x = 2y - 1 Multiply each side by 5.

5x + 1 = 2y Add 1 to each side.

1 _ 2 (5x + 1) = y Divide each side by 2.

5 _ 2 x + 1 _ 2 = y Distribute.

The inverse of f(x) = 2 _ 5 x - 1 _ 5 is f -1(x) = 5 _ 2 x + 1 _ 2 .

xO

f (x) = 2–5x - 1–5

2 4

2

-2

-2

-4

-4

4f ( x )

f –1(x) = 5–2x + 1–2

EXAMPLE

Program: FL MATH REPRINT Component: HANDBOOK1st Pass

Vendor: LASERWORDS Grade: ALGEBRA 2

170 HotTopic 5

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Verifying Inverses

Using composite functions, it is possible to determine whether two given functions are inverses.

If both compositions equal the identity function I(x) = x, then the functions are inverse functions.

Inverse FunctionsTwo functions f(x) and g(x) are inverse functions if and

only if [f ◦ g](x) = x and [g ◦ f](x) = x.

Verify that Two Functions are Inverses

Determine whether the functions are inverses.a. f(x) = 2x - 7 and g(x) = 1 _ 2 (x + 7)

[ f ◦ g](x) = f[ g(x)] [ g ◦ f ](x) = g[ f(x)]

= f [ 1 _ 2 (x + 7)] = g(2x - 7)

= 2 [ 1 _ 2 (x + 7)] - 7 = 1 _ 2 (2x - 7 + 7)

= x + 7 - 7 = x

= x

The functions are inverses since both [ f ◦ g](x) = x and [ g ◦ f ](x) = x.

b. f(x) = 4x + 1 _ 3 and g(x) = 1 _ 4 x - 3

[ f ◦ g](x) = f[ g(x)]

= f ( 1 _ 4 x - 3)

= 4 ( 1 _ 4 x - 3) + 1 _ 3

= x - 12 + 1 _ 3

= x - 11 2 _ 3

Since [ f ◦ g](x) ≠ x, the functions are not inverses.

EXAMPLE

Program: FL MATH REPRINT Component: HANDBOOK1st Pass

Vendor: LASERWORDS Grade: ALGEBRA 2

Inverse Functions and Relations 171

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5•2

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5•2 ExercisesFind the inverse of each function. Then graph the function and its inverse.

1. f(x) = 2 _ 3 x - 1 2. f(x) = 2x - 3

Determine whether each pair of functions are inverse functions. Write yes or no.

3. f(x) = 3x - 1 4. f(x) = 1 _ 4 x + 5

g(x) = 1 _ 3 x + 1 _ 3 g(x) = 4x - 20

5. f(x) = 1 _ 2 x - 10 6. f(x) = 2x + 5

g(x) = 2x + 1 _ 10 g(x) = 5x + 2

7. f(x) = 8x - 12 8. f(x) = -2x + 3

g(x) = 1 _ 8 x + 12 g(x) = - 1 _ 2 x + 3 _ 2

9. f(x) = 4x - 1 _ 2 10. f(x) = 2x - 3 _ 5

g(x) = 1 _ 4 x + 1 _ 8 g(x) = 1 _ 10 (5x + 3)

11. f(x) = 4x + 1 _ 2 12. f(x) = 10 - x _ 2

g(x) = 1 _ 2 x - 3 _ 2 g(x) = 20 - 2x

13. f(x) = 4x - 4 _ 5 14. f(x) = 9 + 3 _ 2 x

g(x) = x _ 4

+ 1 _ 5 g(x) = 2 _ 3 x - 6

15. EXERCISE Alex began a new exercise routine. To gain the maximum benefit from his exercise, Alex calculated his maximum target heart rate using the function f(x) = 0.85(220 - x), where x represents his age. Find the invers e of this function.

Program: FL MATH Component: HANDBOOKPDF Pass

Vendor: LASERWORDS Grade: ALGEBRA 2

172 HotTopic 5

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