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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Functions

xy sin13 xy

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

Each value of x maps to only one value of y . . .

Consider the following graphs

Each value of x maps to only one value of y . . .

BUT many other x values map to that y.

and each y is mapped from only one x.

and

Functions

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

is an example of a one-to-one function

13 xy is an example of a many-to-one function

xy sin

xy sin13 xy

Consider the following graphs

and

Functions

Other one-to-one functions are:

xy

1 xy 4

and

Functions

Here the many-to-one function is two-to-one ( except at one point ! )

432 xxy 863 23 xxxy

Other many-to-one functions are:

This is a many-to-one function even though it is one-to-one in some parts.

It’s always called many-to-one.

Functions

1 xy

This is not a function. Functions cannot be one-to-many.

We’ve had one-to-one and many-to-one functions, so what about one-to-many?One-to-many relationships do exist BUT, by definition, these are not functions.

is one-to-many since it gives 2 values of y for all x values greater than 1.

1,1 xxye.g.

So, for a function, we are sure of the y-value for each value of x. Here we are not sure.

Inverse Functions

SUMMARY 13 xy

xy sin

• A one-to-one function maps each value of x to one value of y and each value of y is mapped from only one x.

e.g. 13 xy

• A many-to-one function maps each x to one y but some y-values will be mapped from more than one x.

e.g.xy sin

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

The 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

Swap x and y:

3

4 x

y

Inverse Functions

The 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:

xy

3

4 x

y

Inverse Functions

The 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:

xy

3

4 x

y

So,3

4)(1 x

xf

Inverse Functions

e.g. 2 Find the inverse function of

1,1

3)(

x

xxf

)(xfNotice that the domain excludes the value of x that would make infinite.

Inverse Functions

e.g. 2 Find the inverse function of

1,1

3)(

x

xxf

Solution:

Let y = the function: 1x

3y

There are 2 ways to rearrange to find x:

Either:

Inverse Functions

Either: 1x3

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:

3y

Inverse Functions

1x3

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

Either:

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

Either:

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

Why are these the same?ANS: x is a common denominator in the 2nd

form

Inverse Functions

So, for 1,1

3)(

x

xxf

x

xxf

xxf

3

)(13

)( 11 or

The domain and range are:

)(0 1 xfx and 1

Inverse Functions

The 1st example we did was for

xxf 34)(

The inverse was 3

4)(1 x

xf

Suppose we form the compound function .

)(1 xff

))(()( 11 xffxff3

)34(4 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.

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

Rearrange:

Swap x and y:

Let

45 xy

xy 54

xy

5

4

yx

5

4

Since the x-term is positive I’m going to work from right to left.

So,5

4)(1 x

xf

Solution: 1. x ,45)( xxf

Inverse Functions

This is an example of a self-inverse function.

Solution: 2. 0x,1

)(x

xf

Letx

y1

Rearrange:y

x1

Swap x and y:x

y1

So, ,1

)(1

xxf 0x

)()(1 xfxf

Inverse Functions

5,5

2)(

x

xxfSolution: 3.

Rearrange:

Swap x and y:

Let 5

2

xy

yx

25

52

y

x

52

x

y

0,52

)(1 xx

xfSo,

Inverse Functions

Solution 4. ,2)( xxf 0x

Rearrange:

Swap x and y:

Let

xy 2

yx 22)2( yx 2)2( xy

So, 21 )2()( xxf

211 )2(2))(()( xxffxff

)2(2 x x

Inverse Functions

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

32)(

xx

xxf

Solution:Rearrange:

Let y = the function:

Multiply by x – 1 :

Careful! We are trying to find x and it appears twice in the equation.

32)1( y x x

1

32

x

xy

The next example is more difficult to rearrange

Inverse Functions

32)1( y x x

Careful! We are trying to find x and it appears twice in the equation.

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

32)(

xx

xxf

Solution:Rearrange: Multiply by x – 1

:

We must get both x-terms on one side.

Let y = the function: 1

32

x

xy

Inverse Functions

x2

3

yy

x 3)2( yy

32 yyx x

32 yyx x

32)1( y x x

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

32)(

xx

xxf

Solution:

Multiply by x – 1 :Remove brackets :Collect x terms on one side:Remove the common factor:Divide by ( y – 2):

Let y = the function: 1

32

x

xy

Rearrange:

Swap x and y:

Inverse Functions

x2

3

yy

x 3)2( yy

32 yyx x

32 yyx x

32)1( y x x

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

32)(

xx

xxf

Solution:

Multiply by x – 1 :Remove brackets :Collect x terms on one side:

Swap x and y:

Remove the common factor:Divide by ( y – 2):

Let y = the function: 1

32

x

xy

Rearrange:

So, ,2

3)(1

x

xxf 2x

2

3

x

yx

Inverse Functions

SUMMARYTo 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

Inverse Functions