example 3 Finding an Inverse Function

Chapter 4.3

a. Find the inverse function of .

b. Graph and its inverse function on the same axes.

2 1( )

3

xf x

2 1( )

3

xf x

2009 PBLPathways

2009 PBLPathways

a. Find the inverse function of .

b. Graph and its inverse function on the same axes.

2 1( )

3

xf x

2 1( )

3

xf x

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

33

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

33

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

33

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

33

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1

2 1

3

2 1

3

3 2 1

3 1 2

3 1

2

3 1( )

2

xy

yx

x y

x y

xy

xf x

Replace f(x) with y

Interchange x and y

Solve for y

Replace y with f -1(x)

33

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2 2 1(2) 1

3f

1 3 1 1(1) 2

2f

2009 PBLPathways

a. Find the inverse function of .2 1

( )3

xf x

1 3 1( )

2

xf x

2 1

( )3

xf x

1. Multiply the input by 2.

2. Subtract 1.

3. Divide by 3.

1. Multiply the input by 3.

2. Add 1.

3. Divide by 2.

2 2 1(2) 1

3f

1 3 1 1(1) 2

2f

2009 PBLPathways

b. Graph and its inverse function on the same axes.2 1

( )3

xf x

2 1( )

3

xf x

2009 PBLPathways

b. Graph and its inverse function on the same axes.2 1

( )3

xf x

2 1( )

3

xf x

1 3 1( )

2

xf x

y x

2009 PBLPathways

b. Graph and its inverse function on the same axes.2 1

( )3

xf x

y x

2009 PBLPathways

b. Graph and its inverse function on the same axes.2 1

( )3

xf x

y x

(2, 1)

2009 PBLPathways

b. Graph and its inverse function on the same axes.2 1

( )3

xf x

y x

(2, 1)

(1, 2)