UNIT 6.7 INVERSE RELATIONSUNIT 6.7 INVERSE RELATIONSAND FUNCTIONSAND FUNCTIONS

Warm UpSolve for x in terms of y.

1.

2.

3.

4. y = 2ln x

Graph and recognize inverses of relations and functions.

Find inverses of functions. Determine whether the inverse of a function is a function.

Write rules for the inverses of functions.

Objectives

inverse functioninverse relationone-to-one function

Vocabulary

You have seen the word inverse used in various ways.

The additive inverse of 3 is –3.

The multiplicative inverse of 5 is

You can also find and apply inverses to relations and functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation.

A relation is a set of ordered pairs. A function is a relation in which each x-value has, at most, one y-value paired with it.

Remember!

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Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation.

Example 1: Graphing Inverse Relations

x 0 1 5 8

y 2 5 6 9

Graph each ordered pair and connect them.

x 2 5 6 9

y 0 1 5 8

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Switch the x- and y-values in each ordered pair.

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Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation.

x 1 3 4 5 6

y 0 1 2 3 5

Check It Out! Example 1

Graph each ordered pair and connect them.

x 0 1 2 3 5

y 1 3 4 5 6 ••

••

•Switch the x- and y-values in each ordered pair.

When the relation is also a function, you can write the inverse of the function f(x) as f–1(x). This notation does not indicate a reciprocal.

Functions that undo each other are inverse functions.

To find the inverse function, use the inverse operation. In the example above, 6 is added to x in f(x), so 6 is subtracted to find f–1(x).

Recall that the vertical-line test (Lesson 1-6) can help you determine whether a relation is a function. Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.

Use the horizontal-line test to determine whether the inverse of the blue relation is a function.

Example 1A: Using the Horizontal-Line Test

The inverse is a function because no horizontal line passes through two points on the graph.

Use the horizontal-line test to determine whether the inverse of the red relation is a function.

Example 1B: Using the Horizontal-Line Test

The inverse is a not a function because a horizontal line passes through more than one point on the graph.

Use the horizontal-line test to determine whether the inverse of each relation is a function.

The inverse is a function because no horizontal line passes through two points on the graph.

Check It Out! Example 1

Recall from Lesson 7-2 that to write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.

Example 2: Writing Rules for inverses

Find the inverse of . Determine whether it is a function, and state its domain and range.

Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.

Example 2 Continued

Rewrite the function using y instead of f(x).

Step 1 Find the inverse.

Simplify.

Switch x and y in the equation.

Cube both sides.

Isolate y.

Example 2 Continued

Because the inverse is a function, .

The domain of the inverse is the range of f(x):{x|x R}.∈

The range is the domain of f(x):{y|y R}.∈

Find the inverse of f(x) = x3 – 2. Determine whether it is a function, and state its domain and range.

Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.

Check It Out! Example 2

Rewrite the function using y instead of f(x).

Step 1 Find the inverse.

Take the cube root of both sides.

Switch x and y in the equation.

Add 2 to both sides of the equation.

Simplify.

Check It Out! Example 2 Continued

y = x3 – 2

x = y3 – 2

x + 2 = y3

3 x + 2 = y

33 3x + 2 = y

The domain of the inverse is the range of f(x): R.

The range is the domain of f(x): R.

Check It Out! Example 2 Continued

Because the inverse is a function, .

You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value.

You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compose two inverses the result is the input value x.

Determine by composition whether each pair of functions are inverses.

Example 3: Determining Whether Functions Are Inverses

Find the composition f(g(x)).

f(g(x)) = 3( x + 1) – 1 1 3

Use the Distributive Property.

Simplify.

f(x) = 3x – 1 and g(x) = x + 1 1 3

Substitute x + 1 for x in f.

1 3

= (x + 3) – 1

= x + 2

Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)).

Example 3 Continued

Check The graphs are not symmetric about the line y = x.

Example 3B: Determining Whether Functions Are Inverses

Find the compositions f(g(x)) and g(f (x)).

For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1 x

1x – 1

Because f(g(x)) = g(f (x)) = x for all x but 0 and 1, f and g are inverses.

= x

= (x – 1) + 1

= x

Example 3B Continued

Check The graphs are symmetric about the line y = x for all x but 0 and 1.

Check It Out! Example 3a

Determine by composition whether each pair of functions are inverses.

Find the composition f(g(x)) and g(f(x)).

3 2

f(x) = x + 6 and g(x) = x – 9 2 3

= x – 6 + 6

= x

f(g(x)) = ( x – 9) + 6 3 2

2 3

g(f(x)) = ( x + 6) – 9 2 3

3 2

= x + 9 – 9

= x

Because f(g(x)) = g(f(x)) = x, they are inverses.

Check The graphs are symmetric about the line y = x for all x.

Check It Out! Example 3a Continued

Find the compositions f(g(x)) and g(f(x)).

Simplify.

Check It Out! Example 3b

Substitute for x in f.

f(g(x)) = + 5

f(x) = x2 + 5 and for x ≥ 0

= x + 25 +5 10 x−

= x – 10 x + 30

Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)).

Check The graphs are not symmetric about the line y = x.

Check It Out! Example 3b Continued

Lesson Quiz: Part I

A: yes; B: no

1. Use the horizontal-line test to determine whether the inverse of each relation is a function.

Lesson Quiz: Part II

D: {x|x ≥ 4}; R: {all Real Numbers}

2. Find the inverse f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.

not a function

3. Determine by composition whether f(x) =

3(x – 1)2 and g(x) = +1 are inverses

for x ≥ 0.

Lesson Quiz: Part III

yes

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