Holt Algebra 2

9-5 Functions and Their Inverses

Holt Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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

33 3x + 2 = y

Holt Algebra 2

9-5 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 Algebra 2

9-5 Functions and Their Inverses

The inverse functions “undo” each other,

You can use composition of functions

to verify that 2 functions are inverses.

When you compose two inverses…the result is the input value of x.

Holt Algebra 2

9-5 Functions and Their Inverses

If f(g(x)) = g(f(x)) = x

Then f(x) and g(x) are inverse functions

Example 1:

f g x g f x x

3f g x x 1

3g f x x

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

Holt Algebra 2

9-5 Functions and Their InversesDetermine by composition whether each pair of functions are inverses.Example 2:

Find the composition f(g(x)).

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

f(g(x)) = x + 2

The functions are NOT inverses.

3 1f g x x

31 1

1 13 3

1x xf

f g x

Holt Algebra 2

9-5 Functions and Their InversesExample 3 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 + 9 – 9

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

26

3f g x x

26

3

3 39 9

2 2xf x

f g x x g f x x

39

2g f x x

39

2

2 26 6

3 3xg x

Holt Algebra 2

9-5 Functions and Their Inverses

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

Simplify.

Example 4

Substitute for x in f.

f(x) = x2 + 5 and for x ≥ 0

= x + 25 +5 10 x

Because f(g(x)) ≠ x, f (x) and g(x) are NOT inverses.

2( ( )) 5f g x x

25 55x xf

10 30f g x x x

Holt Algebra 2

9-5 Functions and Their InversesExample 5: Are the following functions Inverses?

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

1x – 1

Because f(g(x)) = g(f (x)) = x f(x) and g(x) are inverses.

= (x – 1) + 1

gf x x

1( ( ))

1f g x

x

1 1

11

11

x

fx

1

x

1

x

1( ( )) 1g f x

x

11111

1x

x

g

1

1

x

1

1

x

( )g f x x

Holt Algebra 2

9-5 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 Algebra 2

9-5 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 Algebra 2

9-5 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