1. “Teach A Level Maths” Vol. 2: A2 Core Modules
3: Graphs of Inverse Functions
© Christine Crisp

2. Module C3
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3. Consider the graph of the function
The inverse function is

4. Consider the graph of the functionx x x x
The inverse function is
An inverse function is just a rearrangement with x and y swapped.
So the graphs just swap x and y!

5. What else do you notice about the graphs?x x
is a reflection of in the line y = x
The function and its inverse must meet on y = x

6. e.g. On the same axes, sketch the graph of
and its inverse.
N.B!
Solution: x

7. e.g. On the same axes, sketch the graph of
and its inverse.
N.B!
Solution:
N.B. Using the translation of we can see the inverse function is

8. A bit more on domain and range
The previous example used
The domain of is .
Since is found by swapping x and y,
the values of the domain of give the values of the range of
Domain
Range

9. A bit more on domain and range
The previous example used
The domain of is
Since is found by swapping x and y,
the values of the domain of give the values of the range of
Similarly, the values of the range of
give the values of the domain of

10. SUMMARY
The graph of is the reflection of in the line y = x. It follows that the curves meet on y = x
At every point, the x and y coordinates of become the y and x coordinates of
The values of the domain and range of
swap to become the values of the range and domain of
e.g.

11. A Rule for Finding an Inverse
e.g. 1 An earlier example sketched the inverse of the function
There was a reason for giving the domain as
Let’s look at the graph of for all real values of x.

12. x = 1, y = 1 . . . x = 3, y = 1 This function is many-to-one. e.g. and
An inverse function undoes a function.
But we can’t undo y = 1 since x could be 1 or 3.

13. x = 1, y = 1 . . . x = 3, y = 1 This function is many-to-one. e.g. and
An inverse function undoes a function.
But we can’t undo y = 1 since x could be 1 or 3.

14. x = 1, y = 1 . . . x = 3, y = 1 This function is many-to-one. e.g. and
An inverse function undoes a function.
An inverse function only exists if the original function is one-to-one.

15. We can have either
If a function is many-to-one, the domain must be restricted to make it one-to-one.

16. or
If a function is many-to-one, the domain must be restricted to make it one-to-one.

17. e.g. 2 Find possible values of x for which the inverse function of can be defined.
Solution:
Let’s sketch the graph of for
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.

18. e.g. 2 Find possible values of x for which the inverse function of can be defined.
Solution:
Let’s sketch the graph of for
Solution:
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.

19. e.g. 2 Find possible values of x for which the inverse function of can be defined.
Solution:
Let’s sketch the graph of for
Solution:
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.

20. e.g. 2 Find possible values of x for which the inverse function of can be defined.
Solution:
Let’s sketch the graph of for
The function is clearly many-to-one so we must find a domain that gives us a section that is one-to-one.
These values are called the principal values.
In degrees, the P.Vs. are

21. Exercise
Suggest principal values for and
( Give your answers in both degrees and radians )
Solution:

22. Exercise
Suggest principal values for and
( Give your answers in both degrees and radians )
Solution: or

24. or

25. 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

26. Exercise
1 (a) Sketch the function where .
(b) Write down the range of
(c) Suggest a suitable domain for so that the inverse function can be found.
(d) Find and write down its domain and range.
(e) On the same axes sketch

27. Solution:
(a)
(b) Range of :
(c) Restricted domain:
( We’ll look at the other possibility
in a minute. )
(d) Inverse:
Let
Rearrange:
Swap:
Domain:
Range:

28. Solution:
(b) Range of :
(a)
(c)
Suppose you chose
for the domain
(d) Let
As before
Rearrange:
We now need since

29. Solution:
(a)
Range:
(b)
(c)
Suppose you chose
for the domain
Choosing is easier!
(d) Let
As before
Rearrange:
We now need since
Swap:
Domain:
Range:

31. 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.

32. SUMMARY
At every point, the x and y coordinates of become the y and x coordinates of
The values of the domain and range of
swap to become the values of the range and domain of
e.g.
The graph of is the reflection of in the line y = x. It follows that the curves meet on y = x

33. or
For we can have:
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.
either

34. e.g. 1 Find possible values of x for which the inverse function of can be defined.
The function is clearly many-to-one so we must find a domain that gives us a section that is one-to-one.
Let’s sketch the graph of for
Solution:

35. These values are called the principal values.
In degrees, the P.Vs. are
The part closest to the origin is used for the domain.

36. 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