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

2. Module C3
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3. One-to-one and many-to-one functions
Consider the following graphs
and
Each value of x maps to only one value of y . . .
Each value of x maps to only one value of y . . .
and each y is mapped from only one x.
BUT many other x values map to that y.

4. One-to-one and many-to-one functions
Consider the following graphs
and
is an example of a many-to-one function
is an example of a one-to-one function

5. Other one-to-one functions are:
and

6. Other many-to-one functions are:
Here the many-to-one function is two-to-one
( except at one point ! )
This is a many-to-one function even though it is one-to-one in some parts.
It’s always called many-to-one.

7. 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.
e.g.
is one-to-many since it gives 2 values of y for all x values greater than 1.
This is not a function.
Functions cannot be one-to-many.
So, for a function, we are sure of the y-value for each value of x. Here we are not sure.

8. SUMMARY
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.
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.

9. Suppose 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 have
To find y for any x, we have
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.
The notation for the inverse of is

10. Finding an inverse
e.g. 1 For , the flow chart is
Reversing the process:
Notice that we start with x.
The inverse function is
Tip: 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.
Check:
e.g. If

11. 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
Solution:
Let y = the function:
Rearrange ( to find x ):
Swap x and y:

12. 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
Solution:
Let y = the function:
Rearrange ( to find x ):
Swap x and y:

13. 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
Solution:
Let y = the function:
Rearrange ( to find x ):
Swap x and y:
So,

14. e.g. 2 Find the inverse function of
Notice that the domain excludes the value of x that would make infinite.

15. e.g. 2 Find the inverse function of
Solution:
Let y = the function:
There are 2 ways to rearrange to find x:
Either:

16. e.g. 2 Find the inverse function of
Solution:
Let y = the function:
There are 2 ways to rearrange to find x:
Either:

17. e.g. 2 Find the inverse function of
Solution:
Let y = the function:
There are 2 ways to rearrange to find x:
Either:
Swap x and y:

18. e.g. 2 Find the inverse function of
Solution:
Let y = the function:
There are 2 ways to rearrange to find x:
or:
Either:
Swap x and y:

19. e.g. 2 Find the inverse function of
Solution:
Let y = the function:
There are 2 ways to rearrange to find x:
Either:
or:
Swap x and y:
Swap x and y:

20. So, for
Why are these the same?
ANS: x is a common denominator in the 2nd form

21. So, for
The domain and range are:

22. The 1st example we did was for
The inverse was
Suppose we form the compound function
Can you see why this is true for all functions that have an inverse?
ANS: The inverse undoes what the function has done.

23. 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,

24.  Exercise Find the inverses of the following functions: 1.2. 3. 1.  4.
See if you spot something special about the answer to this one.
Also, for this, show

25.  Solution: 1. Let Rearrange:
Since the x-term is positive I’m going to work from right to left.
Swap x and y:
So,

26. Solution: 2.
Let
This is an example of a self-inverse function.
Rearrange:
Swap x and y:
So,

27. Solution: 3.
Let
Rearrange:
Swap x and y:
So,

28. Solution 4.
Let
Rearrange:
Swap x and y:
So,

29. Careful! We are trying to find x and it appears twice in the equation.
e.g. 3 Find the inverse of
Solution:
The next example is more difficult to rearrange
Let y = the function:
Rearrange:
Multiply by x – 1 :
Careful! We are trying to find x and it appears twice in the equation.

30. Careful! We are trying to find x and it appears twice in the equation.
e.g. 3 Find the inverse of
Solution:
Let y = the function:
Rearrange:
Multiply by x – 1 :
Careful! We are trying to find x and it appears twice in the equation.
We must get both x-terms on one side.

31. e.g. 3 Find the inverse of
Solution:
Let y = the function:
Rearrange:
Multiply by x – 1 :
Remove brackets :
Collect x terms on one side:
Remove the common factor:
Divide by ( y – 2):
Swap x and y:

32. e.g. 3 Find the inverse of
Solution:
Let y = the function:
Rearrange:
Multiply by x – 1 :
Remove brackets :
Collect x terms on one side:
Remove the common factor:
Divide by ( y – 2):
Swap x and y:
So,

33. SUMMARY
To find an inverse function:
EITHER:
Step 1: Let y = the function
Step 2: Rearrange ( to find x )
Step 3: Swap x and y
OR:
Write the given function as a flow chart.
Reverse all the steps of the flow chart.

35. 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.

36. is an example of a many-to-one function
One-to-one and many-to-one functions
is an example of a one-to-one function
Consider the following graphs
and

37. e.g. 1 For , the flow chart is
Reversing the process:
Finding an inverse
The inverse function is
Notice that we start with x.
Check:
e.g. If

38. 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
Solution:
Rearrange ( to find x ):
Let y = the function:
Swap x and y:
So,

39. or:
e.g. 2 Find the inverse function of
There are 2 ways to rearrange to find x:
Solution:
Let y = the function:
Swap x and y:
Either:

40. So, for x f + = - 3 ) ( 1 or

41. e.g. 3 Find the inverse of
Solution:
Rearrange:
Multiply by x – 1 :
Remove brackets :
Collect x terms on one side:
Remove the common factor:
Swap x and y:
Divide by ( y – 2):
So,
Let y = the function:

42. SUMMARY
To 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