2: Inverse Functions2: Inverse Functions

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

Module C3

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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

Function Inverse Function

x2

xa

+

-

reciprocate

x1

ax

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

f–1(x) = 1x

So the inverse of is 1x

1x

f(x) = and f–1(x) = 1x

1x

4

3

2

11

f(x)x

121314

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) =

Domain x 0 as you cannot a negative number

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 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)

5x2

5x5

2x

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

Consider

1,1

3)(

x

xxf

x

xxf

xxf

3

)(13

)( 11 or

Why are these the same?ANS: add up the fractions

xxf

13

)(1 3 1

1x

3

1

x

x

3 x

x

An alternative Answer

Cross and Smile

Inverse Functions

Ex.2 f:x x find f -1(x)25

3 4x

1 2 4

3 x 5 1

f –1 (x)

1 2 4 20

3 5

x

x

1 4 18

3 5

x

x

Done earlier

Cross and Smile 1 2 4

3 x 5 1

Inverse Functions

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 xxf 34)(

The inverse is 3

41

xxf )(

x -3 +4 f(x)

inverse of -3 is

3

4

x3

4 x

-3

Inverse Functions

The previous example was for

xxf 34 )(

The inverse was 3

41 xxf

)(

Suppose we form the compound function . )(1 xff

3

344 )( x

3

344 x

x)(1 xff Can you see why this is true for all functions that have an inverse?

ANS: The inverse undoes what the function has done.

f–1(4 – 3x) ))(()( xffxff 11

Inverse Functions

xxffxff )()( 11

The order in which we find the compound function of a function and its inverse makes no difference.For all functions which have an inverse,

)( xf

Inverse FunctionsExercise

Find the inverses of the following functions:

,2)( xxf 0x

2.

3. 5,5

2)(

x

xxf

,45)( xxf1. x

,1

)(x

xf 0x

4.

See if you spot something special about the answer to this one.

Also, for this, show

xxff )(1

Inverse Functions

So,5

4)(1 x

xf

Solution: 1. x ,45)( xxf

Solution: 2. 0x,1

)(x

xf

So, ,1

)(1

xxf 0x

5,5

2)(

x

xxfSolution: 3.

0,52

)(1 xx

xfSo,

Solution 4. ,2)( xxf 0xSo, 21 )2()( xxf

Inverse Functions

Using Long Division to Find Inverses

As x appears in 2 places it is impossible to go forwards

and backwards using the order of operations.

Do long division.x 2 x

So now x appears in

only 1 place.

Ex.1 f:x x-2x

x 2

x + 2 x

1

x+2

-2

x+2

2–

Inverse Functions

f: x2

1-x+2

x

22

x 1

f –1(x) = 2

x 1

x 1

2

x – 1

22

x 1 f –1(x) =

2 2x 1 1

2 2x 2x 1

2x x 1

x 1

List the operations in the order applied

Go backwards finding the inverse of each operation

Simplify f–1

(x)

+2 recip -2 +1 f(x)

Inverse Functions

SUMMARYTo find an inverse

function:

•Write the given function as a flow chart.

•Reverse all the steps of the flow chart.

Inverse Functions

Inverse Functions

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Inverse Functions

is an example of a many-to-one function

xy sin

One-to-one and many-to-one functions

is an example of a one-to-one function

13 xy

xy sin13 xy

Consider the following graphs

and

Inverse Functions

x 2 4x2 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

xf

Notice that we start with x.

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

)(1f 14 52

414 4)5(2

Inverse FunctionsThe flow chart method of finding an inverse

can be slow and it doesn’t always work so we’ll now use another method.

e.g. 1 Find the inverse of xxf 34)( Solution:

xy 34 Rearrange ( to find x ):

Let y = the function:

yx 43

3

4

Swap x and y:

x y

3

4 xy

So,3

4)(1 x

xf

Inverse Functions

or: )1( xy 31x

3 y

e.g. 2 Find the inverse function of

1,1

3)(

x

xxf

1xThere are 2 ways to rearrange to find x:

Solution:

Let y = the function:

Swap x and y: 13

x

y

13

y

x

3y

Swap x and y: x

xy

3

3 yyxyyx 3

y

yx

3

Either:

Inverse Functions

So, for 1,1

3)(

x

xxf

x

xxf

xxf

3

)(13

)( 11 or

Inverse Functions

e.g. 3 Find the inverse of 1,1

32)(

xx

xxf

Solution:Rearrange: 32)1( yMultiply by x – 1

:Remove brackets :

32 yyCollect x terms on one side: 32 yyRemove the common factor: 3)2( yy

x x

1

32

x

xy

x x

x x

x

Swap x and y:

Divide by ( y – 2):2

3

x yy

So, ,2

3)(1

x

xxf 2x

2

3

x

yx

Let y = the function:

Inverse FunctionsSUMMAR

YTo find an inverse function:

EITHER:

•Write the given function as a flow chart.

•Reverse all the steps of the flow chart.

OR:

•Step 2: Rearrange ( to find x )

•Step 1: Let y = the function

•Step 3: Swap x and y