Finding the Inverse of a Function : The procedure for finding the inverse of a function is a rather simple one although there are a couple of steps which can on occasion be somewhat messy. Following is the process
Given the function f ( x ) we desire to determine the inverse function, f -1 ( x ) .
a. Firstly, replace f ( x ) with y. It is done to make the rest of the procedure easier.
b. Replace every x to a y and replace every y to an x.
c. Solve out the equation from Step 2 for y. it is the step where mistakes are most frequently made so be careful with this step.
d. Replace y with f -1 ( x ) . In other terms, we've managed to determine the inverse at this point!
e. Check your work by verifying that (f o f -1 )( x ) = x and ( f -1 + f )( x ) = x are both true. Sometimes this work can be messy making it simple to make mistakes so again be careful.
That's the procedure. Most of the steps are not all that bad although as mentioned in the procedure there are a couple of steps that we actually have to be careful with as it is easy to commit mistakes in those steps.
In the verification step technically we really do have to check that both ( f o f -1 )( x ) = x and ( f -1 o f )( x ) = x are true. For all of the functions which we are going to be looking at in this course if one is true then the other will also be true. Though, there are functions for which it is probable for only one of these to be true. It is brought up since in all the problems here we will be just verifying one of them. We just have to always remember that technically we have to check both.