Section 4-5 Inverse Functions

Objective: To find the inverse of a function, if the inverse

exists.

FunctionsImagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye, B(x), and the result is a blue egg (y).

The Inverse Function “undoes” what the function does.

The Inverse Function of the Blue dye is bleach.

The bleach will “undye” the blue egg and make it white.

In the same way, the inverse of a given function will “undo” what the original function did.

For example, let’s take a look at the square function: f(x) = x2

3

x f(x)

33333 9999999

y 𝒇−𝟏(𝒙)

9999999 3333333

x2 x

555555 252525

252525

252525255 55555555

In the same way, the inverse of a given function will “undo” what the original function did.

For example, let’s take a look at the square function: f(x) = x2

x f(x) y 𝒇−𝟏(𝒙)

x2 x

111111111111 121121121121121121121121121121121121121121 1111111111111111

In the same way, the inverse of a given function will “undo” what the original function did.

For example, let’s take a look at the square function: f(x) = x2

x f(x) y 𝒇−𝟏(𝒙)

x2 x

Inverse Function Definition

Two functions f and g are called inverse functions if the following two statements are true:

1. 𝑔(𝑓 𝑥 ) = 𝑥 for all x in the domain of f.

2. 𝑓(𝑔 𝑥 ) = 𝑥 for all x in the domain of g.

Graphically, the x and y values of a point are switched.

The point (4, 7)

has an inverse point of (7, 4)

AND

The point (-5, 3)

has an inverse point of (3, -5)

Graphically, the x and y values of a point are switched.

If the function y = g(x) contains the points

then its inverse, y = g-1(x), contains the points

x 0 1 2 3 4

y 1 2 4 8 16

x 1 2 4 8 16

y 0 1 2 3 4

Where is there a line of reflection?

The graph of a function and its

inverse are mirror images about the line

𝒚 = 𝒙𝒚 = 𝒇(𝒙)

𝒚 = 𝒇−𝟏(𝒙)

y = x

Inverse Notes

•The inverse of a function f is written𝑓−1 and is read “f inverse”

• 𝑓−1(𝑥) is read, “f inverse of x”• If point (𝑥, 𝑦) is on graph of f, then point (𝑦, 𝑥) is on the graph of the inverse of f.

• The graph of 𝒇−𝟏is the reflection of the graph of 𝒇 in the line 𝑦 = 𝑥

Find the inverse of a function algebraically:

Example 1: f(x) = 6x - 12

Step 1: Switch x and y

x = 6y - 12

Step 2: Solve for y

x 6y 12

x 12 6y

x 12

6 y

1

6x 2 y

*Note: You can replace f(x) with y.

𝒇−𝟏 𝒙 =𝟏

𝟔𝒙 + 𝟐

Given the function: f(x) = 3x2 + 2 Find the inverse.

Step 1: Switch x and yx = 3y2 + 2

Step 2: Solve for yx 3y2 2

x 2 3y2

x 2

3 y2

x 2

3 y

Example 2:

𝒇−𝟏 𝒙 =𝒙 − 𝟐

𝟑

On the same axes, sketch the graph of

and its inverse.

2,)2(2 xxy

Notice

)0,2(

)1,3(

xy

)4,4(x

Solution:

)2,0(

)3,1(

On the same axes, sketch the graph of

and its inverse.

2,)2(2 xxy

Noticexy

2)2( xy

Solution:

Using the translation of what is the equation of the inverse function?

x

2 xy

2)(1

xxf

2)2( xy

2 xy

domain and range

The domain of is . 2x

)(xf

Since is found by swapping x and y,

)(1

xf

2)2()( xxf 2xDomain

2y2)(1

xxf Range

2,)2()(2 xxxfThe previous example used .

the values of the domainof give the values of the range of .

)(xf)(

1xf

2)2( xy

domain and range

2,)2()(2 xxxfThe previous example used .

The domain of is . 2x

)(xf

2 xySince is found by swapping x and y,

)(1

xf

)(1

xfgive the values of the domain of

the values of the domainof give the values of the range of .

)(xf)(

1xf

Similarly, the values of the range of )(xf