## Inverse Functions

Inverse functions are only defined if the original function is a one to one function, which means that every point on the x axis only corresponds to one point on the y axis. In fact we have already seen an inverse function in our exponential and log functions.

The inverse function can be found by switching around x and y (f(x)).

Take logs of both sides

Here is the graph of the two functions:

The inverse is a reflection of the original function in the black line y=x.

When the function is flipped to the inverse function the new domain becomes the old range and the old domain becomes the new range.

- Dom f
^{-1}=ran f
- Ran f
^{-1}=dom f

## Example

2014 exam 2

We know that there can only be an inverse function if the function is a one to one function.

For f(x):

If a cubic function has a point of inflection then it will be a one to one function, therefore our first step is to find the derivative.

Now we solve for x in f'(x)=0.

Therefore we have two turning points and it is not a one to one function, so we need to restrict the domein.

There are three sections of the graph where it is a one to one function

- D=(∞
^{–}, -3)
- D=( -3,7)
- D=(7,∞
^{+})

Only one of those are the same as the above which is the D:(∞^{–}, -3) which is A in the multiple choice.

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