Inverse FunctionsInverse Functions

© Christine Crisp

Inverse Functions

42 xySuppose we want to find the value of y when x = 3 if

We can easily see the answer is 10 but let’s write out the steps using a flow chart.

We haveTo find y for any x, we have

3 6 10

To find x for any y value, we reverse the process. The reverse function “undoes” the effect of the original and is called the inverse function.

2 4

x 2 4x2 42 x y

The notation for the inverse of is)( xf )(1 xf

Inverse Functions

2 4x x2 42 x

42)( xxfe.g. 1 For , the flow chart is

2

4x 2 4x x4

Reversing the process:

Finding an inverse

The inverse function is 2

4)(1

x

xfTip: A useful check on the working is to substitute any number into the original function and calculate y. Then substitute this new value into the inverse. It should give the original number.

Notice that we start with x.

Check:

52

414 4)5(2

)(1f 14

14e.g. If ,5x 5 )(f

Inverse Functions

x1

ax i.e flip power

Remember the inverse function performs the reverse effect

-

+

Inverse Functions

Using the Reciprocal Function

Ex.1 f(x)= find f–1 (x)1x

To find the inverse we need a function which will change

½ back into 2 and ¼ back into 4 etc

So the inverse of is 1x

1x

f(x) = and f–1(x) = 1x

1x

4

3

2

11

f(x)x

121314

12

12

13

13

14

14

Inverse Functions

Function Inverse Function

x2

xa

+

-

reciprocate

x1

ax

Remember the inverse function performs the reverse effect

-

+

reciprocate

Inverse Functions

Finding the inverse of a function

Ex.1 f:x= 2(x+3)2 find f–1 (x)

List the operations in the order applied

x To find the inverse go backwards finding the inverse of each operation

x

so f –1 (x) = x

32

+3 square x 2 f(x)

2 square root -3f –1 (x)

2x

2x

32

x

Inverse Functions

As the original x value is obtained the inverse function is correct

The result can be checked by substitution

so f(2) =

substitute this value into the inverse function f-1(x)

f-1(50) =50

3 25 3 22

f(x)= 2(x+3)2 2(2+3)2 = 50

Inverse Functions

x

Ex.2 f:x find f -1(x)

f(x)

25

3 4x

x

1 24

3 x 5

f –1 (x)

List the operations in the order applied

Go backwards finding the inverse of each operation

3 -4 reciprocate 2 +5

-5 2 reciprocate + 4 3f–1(x)

5x25x

52x

45

2 x

4

52

31

x

Inverse Functions

Checking f(2) =

Substitute x = 6 into f–1(x)

f –1 (6)

2f x 5

3x 4

( )

25 6

3 2 4

This is the original x value.

The result can be checked by substitution

1 2

43 6 5

1 24

3 x 5

=2

Inverse Functions

x

Ex.3 f:x find f -1(x)

f(x)

3

22 5x

x

f –1 (x)=

List the operations in the order applied

Go backwards finding the inverse of each operation

Power 2 -5

+5 2 Power f–1(x)

5x 5

2

x 2

35

2

x

f–1(x)

2

35

2

x

Inverse Functions

Ex 4 Changing the Sign

Ex.1 f:x 5 - x

To change the sign of x multiply by –1

x -1 +5 f(x)

f–1(x) -1 -5 x

inverse of -1 is

f–1(x) = (x 5) x 5 5 x

Which is the same as -1

-1

Inverse Functions

Ex 5 xxf 34)(

The inverse is 3

41

xxf )(

x -3 +4 f(x)

inverse of -3 is

3

4

x3

4 x

-3

Inverse FunctionsExercise

Find the inverses of the following functions:

,2)( xxf 0x

2.

3.2

( )6

f xx

,45)( xxf1.

1( ) ,

5f x

x

4.

Inverse Functions

So,5

4)(1

x

xf

Solution: 1. ,45)( xxf

Solution: 2.1

( ) ,5

f xx

So, 1 1( ) 5,f x

x

2( )

6f x

x

Solution: 3.

1 2( ) 6 ,f x

x So

,

Solution 4. ,2)( xxf So, 21 )2()( xxf

Inverse Functions

SUMMARYTo find an inverse

function:

•Write the given function as a flow chart.

•Reverse all the steps of the flow chart.