If we are given a formula for a 1-1 function as y=f(x) then there is an algebraic method that attempts to find a formula for . Whether or not the method is successful usually depends on whether or not the algebraic steps can be carried out. The basic idea is this. Since a point (x,y) in the graph of f satisfies the equation y=f(x), the point (y,x) in the graph of should satisfy the equation x=f(y). The only problem is can you solve this for y. That just means can you algebraically solve x=f(y) for y in terms of x. If you can, you will have , the formula for the inverse function. So the steps are:
(1) write down y=f(x)
(2) switch x and y to get x=f(y)
(3) solve for y to get
Here's a simple example: Let a 1-1 and onto function from R to R. Here are the steps.
(3)solving for y:
So we get .