“Teach A Level Maths”
Vol. 2: A2 Core Modules
3: Graphs of Inverse
Functions
© Christine Crisp
Module C3
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Inverse Functions
Consider the graph of the function
f ( x)  2 x  4
y  2x  4
x4
y
2
The inverse function is
f
1
x4
( x) 
2
Inverse Functions
Consider the graph of the function
y  2x  4
( 3,  2)x
x (0, 4)
(4, 0)
x
x4
y
2
x
( 2,  3)
The inverse function is
f ( x)  2 x  4
f
1
x4
( x) 
2
An inverse function is just a rearrangement with x
and y swapped.
So the graphs just swap x and y!
Inverse Functions
What else do you notice about the graphs?
y  2 x  4x (0, 4)
yx
(4, 0)
x
( 3,  2)x
( 4,  4x)
x
x4
y
2
( 2,  3)
f 1 ( x ) is a reflection of f ( x ) in the line y = x
The function and its inverse must meet on y = x
Inverse Functions
e.g. On the same axes, sketch the graph of
y  ( x  2) ,
x2
2
and its inverse.
Solution:
N.B!
yx
(4, 4)
x
(1, 3)
(0, 2)
( 3, 1)
( 2, 0)
Inverse Functions
e.g. On the same axes, sketch the graph of
y  ( x  2) ,
2
x2
and its inverse.
Solution:
N.B!
yx
y x 2
y  ( x  2) 2
N.B. Using the translation of x
inverse function is f 1 ( x ) 
we can see the
x 2 .
Inverse Functions
A bit more on domain and range
The previous example used
y  x 2
y  ( x  2) 2
f ( x )  ( x  2) 2 , x  2 .
The domain of f ( x ) is
x2.
1
Since f ( x ) is found
by swapping x and y,
the values of the domain
of f ( x ) give the values of
1
the range of f ( x ).
f ( x )  ( x  2) 2 Domain x  2
1
f ( x )  x  2 Range y  2
Inverse Functions
A bit more on domain and range
The previous example used
y  x 2
y  ( x  2) 2
f ( x )  ( x  2) 2 , x  2 .
The domain of f ( x ) is
x2 .
1
Since f ( x ) is found
by swapping x and y,
the values of the domain
of f ( x ) give the values of
1
the range of f ( x ).
Similarly, the values of the range of f ( x )
give the values of the domain of
f 1 ( x )
Inverse Functions
SUMMARY
1
• The graph of y  f ( x ) is the reflection of y  f ( x )
in the line y = x. It follows that the curves meet on
y=x
•
At every point, the x and y coordinates of y  f ( x )
become the y and x coordinates of y  f 1 ( x ) .
•
The values of the domain and range of f ( x )
swap to become the values of the range and
domain of f 1 ( x ).
y  x2  2
e.g.
f ( x)  x  2 ,
1
f ( x)  x  2 ,
2
x2 ;
x  0;
y 0
y 2
yx
y  x2
Inverse Functions
A Rule for Finding an Inverse
e.g. 1 An earlier example sketched the inverse of the
function
y  ( x  2) 2 ,
x2
There was a reason for giving the domain as
Let’s look at the graph of
values of x.
x2.
y  ( x  2) 2 for all real
Inverse Functions
y  ( x  2) 2
This function is
many-to-one.
e.g.
x = 1, y = 1 . . .
and
x = 3, y = 1
An inverse function undoes a function.
But we can’t undo y = 1 since x could be 1 or 3.
Inverse Functions
y  ( x  2) 2
This function is
many-to-one.
e.g.
x = 1, y = 1 . . .
and
x = 3, y = 1
An inverse function undoes a function.
But we can’t undo y = 1 since x could be 1 or 3.
Inverse Functions
y  ( x  2) 2
This function is
many-to-one.
e.g.
x = 1, y = 1 . . .
and
x = 3, y = 1
An inverse function undoes a function.
An inverse function only exists if the original
function is one-to-one.
Inverse Functions
y  ( x  2) 2
We can have either
x2
If a function is many-to-one, the domain must
be restricted to make it one-to-one.
Inverse Functions
y  ( x  2) 2
or
x2
If a function is many-to-one, the domain must
be restricted to make it one-to-one.
Inverse Functions
e.g. 2 Find possible values of x for which the
inverse function of sin x can be defined.
Solution: Let’s sketch the graph of y  sin x for
 2  x  2
The function is
clearly many-to-one
so we must find a
domain that gives us
a section that is
one-to-one.
The most obvious section to use is the part close to
the origin.
Inverse Functions
e.g. 2 Find possible values of x for which the
inverse function of sin x can be defined.
Solution: Let’s sketch the graph of y  sin x for
 2  x  2
The function is
clearly many-to-one
so we must find a
domain that gives us
a section that is
one-to-one.
The most obvious section to use is the part close to
the origin.
Inverse Functions
e.g. 2 Find possible values of x for which the
inverse function of sin x can be defined.
Solution: Let’s sketch the graph of y  sin x for
 2  x  2
The function is
clearly many-to-one
so we must find a
domain that gives us
a section that is
one-to-one.
The most obvious section to use is the part close to
the origin.
Inverse Functions
e.g. 2 Find possible values of x for which the
inverse function of sin x can be defined.
Solution: Let’s sketch the graph of y  sin x for
 2  x  2
The function is
clearly many-to-one
so we must find a
domain that gives us
a section that is
one-to-one.


  x  These values are called the principal values.
2
2
In degrees, the P.Vs. are  90   x  90 
Inverse Functions
Exercise
Suggest principal values for y  cos x and y  tan x
( Give your answers in both degrees and radians )
Solution:
y  cos x
Inverse Functions
Exercise
Suggest principal values for y  cos x and y  tan x
( Give your answers in both degrees and radians )
Solution:
y  cos x
0 x  
or
0  x  180


y  tan x
Inverse Functions
y  tan x
Inverse Functions


  x 
2
2
or
 90   x  90 
Inverse Functions
SUMMARY
•
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.
radians
degrees
sin x

2
x

2
0 x  
cos x
tan x



2
x
 90   x  90 
0   x  180 

2
 90  x  90


Inverse Functions
Exercise
1 (a) Sketch the function y  f ( x ) where
f ( x)  x 2  1 .
(b) Write down the range of f ( x ) .
(c) Suggest a suitable domain for f ( x ) so that
the inverse function f 1 ( x ) can be found.
(d) Find f
range.
1
( x ) and write down its domain and
(e) On the same axes sketch y  f
1
( x) .
Inverse Functions
Solution:
y  x2  1
(a)
(b) Range of f ( x ) :
f ( x )  1
(c) Restricted domain:
x0
(d) Inverse: Let
Rearrange:
Swap:

Domain:
y  x2  1
y  1  x2
( We’ll look at the
other possibility
x0
in a minute. )
y 1  x
x 1  y
f 1 ( x )  x  1
x  1
Range: y  0
Solution:
(a)
Inverse Functions
(b) Range of f ( x ) :
y  x2  1
f ( x )  1
(c) Suppose you chose
x0
for the domain
(d)
As before
Rearrange:
We now need
y  x2  1
y  1  x2
 y  1  x since x  0
Let
Solution:
(a)
Inverse Functions
y  x2  1
(b) Range: y  1
(c) Suppose you chose
x0
(d)
As before
Rearrange:
Let
for the domain
Choosing x  0
2
y  x 1
is easier!
y  1  x2
We now need  y  1  x since x  0
Swap:
 x 1  y
f 1 ( x )   x  1
Domain: x  1 Range: y  0
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
SUMMARY
1
• The graph of y  f ( x ) is the reflection of y  f ( x )
in the line y = x. It follows that the curves meet on
y=x
•
At every point, the x and y coordinates of y  f ( x )
become the y and x coordinates of y  f 1 ( x ) .
•
The values of the domain and range of f ( x )
swap to become the values of the range and
domain of f 1 ( x ).
y  x2  2
e.g. f ( x ) 
x2,
x2 ;
y 0
yx
f 1 ( x )  x 2  2 ,
x  0;
y 2
y  x2
Inverse Functions
•
An inverse function undoes a function.
•
An inverse function only exists if the original
function is one-to-one.
•
If a function is many-to-one, the domain must
be restricted to make it one-to-one.
For y  ( x  2) 2 we can have:
either
x2
y  ( x  2) 2
or
x2
y  ( x  2) 2
Inverse Functions
e.g. 1 Find possible values of x for which the
inverse function of sin x can be defined.
Solution:
Let’s sketch the graph of y  sin x for
 2  x  2
y  sin x
The function is
clearly many-to-one
so we must find a
domain that gives us
a section that is
one-to-one.
Inverse Functions
y  sin x
The part closest to
the origin is used for
the domain.


  x 
2
2
These values are called the principal values.
In degrees, the P.Vs. are  90  x  90
SUMMARY
Inverse Functions
•
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.
radians
degrees
sin x

2
x

2
0 x  
cos x
tan x



2
x
 90   x  90 
0   x  180 

2
 90  x  90

