Holt McDougal Algebra 2

6-6 Functions and Their Inverses 6-6 Functions and Their Inverses

Holt Algebra2

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Algebra 2

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

Warm Up Solve for x in terms of y.

1.

2.

3.

4. y = 2ln x

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

Determine whether the inverse of a function is a function.

Write rules for the inverses of functions.

Objectives

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

one-to-one function Vocabulary

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

In Lesson 7-2, you learned that the inverse of a function f(x) “undoes” f(x). Its graph is a reflection across line y = x. The inverse may or not be a function.

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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses 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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses 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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses 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}. ∈

Check Graph both relations to see that they are symmetric about y = x.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

3 3 3 x + 2 = y

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

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

Check Graph both relations to see that they are symmetric about y = x.

Check It Out! Example 2 Continued

Because the inverse is a function, .

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses 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 1

x – 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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses Example 3B Continued

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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses 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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

Check It Out! Example 3a Continued

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses 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.

Holt McDougal Algebra 2

6-6 Functions and Their Inverses 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

Holt McDougal Algebra 2

6-6 Functions and Their Inverses

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