graphs of inverse functions (1)

8/10/2019 Graphs of Inverse Functions (1)

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3: Graphs of Inverse

Functions

Christine Crisp

Teach A Level Maths

Vol. 2: A2 Core Modules


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Module C3

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with

permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"


3/36

Inverse Functions

42 xy

Consider the graph of the function 42)( xxf

The inverse function is24)(

1

xxf

42 xy

24 xy
http://localhost/var/www/apps/conversion/tmp/scratch_8/0%20Contents.ppt


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Inverse Functions

42 xy

24 xy

Consider the graph of the function 42)( xxf

The inverse function is24)(

1

xxf

An inverse function is just a rearrangement with xand yswapped.

So the graphs just swap xand y!

)4,0(x

)0,4(x

)2,3( x

)3,2( x


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Inverse Functions

42 xy

2

4

xy

)4,0(x

)0,4(

x

)2,3( x

)3,2( x

is a reflection of in the line y = x)(1 xf )(xf

xy

What else do you notice about the graphs?

)4,4( x

The function and its inverse must meet ony = x


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Inverse Functions

e.g. On the same axes, sketch the graph of

and its inverse.

2,)2( 2

xxy

N.B!

)0,2(

)1,3(

xy

)4,4(

x

Solution:

)2,0(

)3,1(


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Inverse Functions

e.g. On the same axes, sketch the graph of

and its inverse.

2,)2( 2

xxy

N.B!

xy

2)2( xy

Solution:

N.B. Using the translation of we can see theinverse function is .

x2)(

1xxf

2xy


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Inverse Functions

2)2(

xy

2xy

A bit more on domain and range

The domainof is.2x

)(xf

Since is foundby swapping xand y,

)(1

xf

2)2()( xxf 2xDomain

2y2)(1

xxf Range

2,)2()(

2

xxxfThe previous example used .

the values of the domainof givethe values of

the rangeof .

)(xf

)(

1

xf


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Inverse Functions

2xy

xy2

2xy

2xy

SUMMARY

The graph of is the reflection of

in the line y = x. It follows that the curves meet ony = x

)(xfy

)(1

xfy

At every point, the xand ycoordinates ofbecome the yand xcoordinates of .

)(xfy

)(1

xfy

The values of the domain and range ofswap to become the values of the range anddomain of .

)(xf

)(1

xf

yxxxf ;,2)( 21

20

e.g. yxxxf ;,2)( 2 0


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Inverse Functions

A Rule for Finding an Inverse

e.g. 1 An earlier example sketched the inverse of thefunction

2,)2( 2

xxy

There was a reason for giving the domain as .2x

Lets look at the graph of for allrealvalues of x.

2)2( xy


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Inverse Functions

2)2(

xy

This function is

many-to-one.e.g.

x= 1, y= 1. . .

andx= 3, y= 1

An inverse function undoes a function.

But we cant undo y= 1 since xcould be 1or 3.


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Inverse Functions

2)2(

xy

This function is

many-to-one.e.g.

x= 1, y= 1. . .

andx= 3, y= 1


But we cant undo y= 1 since xcould be 1or 3.


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Inverse Functions

2)2(

xy

This function is

many-to-one.e.g.

x= 1, y= 1. . .

andx= 3, y= 1


An inverse function only exists if the originalfunction is one-to-one.


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Inverse Functions

2)2(

xy

If a function is many-to-one, the domain mustbe restricted to make it one-to-one.

We can have either2x


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Inverse Functions


or2x

2)2(

xy


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Inverse Functions

e.g. 2 Find possible values of xfor which theinverse function of can be defined.xsin

Solution: xy sinLets sketch the graph of for22

x

The most obvious section to use is the part close tothe origin.

The function isclearly many-to-one

so we must find adomain that gives usa section that isone-to-one.


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Inverse Functions




Solution:Solution: xy sinLets sketch the graph of for22

x



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Inverse Functions




Solution:Solution: xy sinLets sketch the graph of for22

x



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Inverse Functions



These values are called the principalvalues.

In degrees, the P.Vs. are 9090 x

22

x

Solution: xy sinLets sketch the graph of for22

x



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Inverse Functions

( Give your answers in both degrees and radians )

Exercise

Suggest principal values for andxy cos xy tan

Solution:

xy cos


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Inverse Functions

( Give your answers in both degrees and radians )

Exercise

Suggest principal values for andxy cos xy tan

Solution:

x0

1800 x

or

xy cos


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Inverse Functionsxy tan


24/36

Inverse Functionsxy tan

22

x

9090 x

or


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Inverse FunctionsSUMMARY

Only one-to-one functions have an inverse

function. If a function is many-to-one, the domain must

be restricted to make the function one-to-one.

The restricted domains of the trig functions

are called the principal values.

22

x

radians

9090 x

degrees

xcos x0 1800 x

xsin

xtan22

x

9090 x

F


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Inverse FunctionsExercise

(d) Find and write down its domain andrange.

1 (a) Sketch the function where

.

)(xfy

(e) On the same axes sketch .

)(1

xf

1)( 2

xxf

)(xf

)(1

xf

)(1

xfy

(c) Suggest a suitable domain for so thatthe inverse function can be found.

(b) Write down the range of .)(xf

I F i


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Inverse Functions

12xy(a)

Solution:

0x( Well look at the

other possibilityin a minute. )

Rearrange: 21 xy

xy 1

Swap: yx 1

Let 12xy(d) Inverse:

1)(1

xxf

Domain: 1x Range: 0y

0x(c) Restricted domain:

(b) Range of :)(xf1)(

xf

I F i


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Inverse Functions

12xy

Solution:

(a)

Rearrange: 21 xy

(d) Let 12

xyAs before

(c)0x

Suppose you chose

for the domain

We now need sincexy 1 0x

(b) Range of :)(xf1)(

xf

I F ti


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Inverse Functions

12xy

Solution:

(a)

Swap: yx 1

1)(1

xxf

Range:(b) 1y

Domain: 1x Range: 0y

(c)0x

Suppose you chose

for the domain

Rearrange: 21 xy

(d) Let 12

xyAs before

We now need sincexy 1 0x

Choosingis easier!

0x

I F ti


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Inverse Functions

I F ti


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Inverse Functions

The following slides contain repeats ofinformation on earlier slides, shown withoutcolour, so that they can be printed and

photocopied.

For most purposes the slides can be printedas Handouts with up to 6slides per sheet.


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Inverse Functions

xy

22xy

SUMMARY

At every point, the xand ycoordinates ofbecome the yand xcoordinates of .

)(xfy

)(1

xfy

The values of the domain and range ofswap to become the values of the range anddomain of .

)(xf

)(1

xf

yxxxf ;,2)( 21

20

e.g. yxxxf ;,2)( 2 0

2xy

The graph of is the reflection of

in the line y = x. It follows that the curves meet ony = x

)(xfy

)(1

xfy


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Inverse Functions

or 2xFor we can have:

2x


An inverse function only exists if the originalfunction is one-to-one.


2)2( xy2)2( xy

2

)2( xyeither


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Inverse Functions


The function isclearly many-to-oneso we must find adomain that gives usa section that is

one-to-one.

xy sinLets sketch the graph of for

22 x

Solution:

xy sin


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Inverse Functions

22

x

These values are called the principal values.

In degrees, the P.Vs. are

The part closest tothe origin is used forthe domain.

xy sin

9090 x

F


36/36

Inverse FunctionsSUMMARY

Only one-to-one functions have an inverse

function. If a function is many-to-one, the domain must

be restricted to make the function one-to-one.

The restricted domains of the trig functionsare called the principal values.

22

x

radians

9090 x

degrees

xcos x0 1800 x

xsin

xtan

22

x

9090 x

graphs of inverse functions (1)

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