The process for finding the inverse of a function is a quite 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 ).
1. First, replace f ( x ) with y. It is done to make the rest of the procedure easier.
2. Replace each x with a y & replace each y along with an x.
3. Solve out the equation through Step 2 for y. It is the step where mistakes are most frequently made so be careful along with this step.
4. Replace y with f ^{-1} ( x ) . In other terms, we've managed to determine the inverse at this point!
5. Check your work by verifying that ( f o f ^{-1} )( x ) ? x and ( f ^{-1} o f )( x ) = x are both true. This work sometimes can be messy making it easy to commit mistakes so again be careful.
That's the procedure. Mostly steps are not all that bad but as specified in the procedure there are a couple of steps that we actually need to be careful with.
In the verification step technically we really do need to check that both ( f of ^{-1 })( x ) = x and
( f^{ -1} o f )( x )= x are true. For all the functions which we are going to be looking at in this section if one is true then the other will also be true. Though, there are functions for which it is possible for only of these to be true. It is brought up since in all the problems here we will be just checking one of them. We only need to always remember that we should technically check both.