1. 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

2. 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

3. 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:

4. 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:

5. 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,

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

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

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

9. 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:

10. 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:

11. 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:

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

13. So, for
The domain and range are:

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

15. The order in which we find the composite function of a function and its inverse makes no difference.
For all functions which have an inverse,

16.  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

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

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

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

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

21. 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.

22. 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.

23. 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:

24. 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,

25. 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.