To derive the form of the inverse function, take your given function in the form of . The switch the x and y variables to get and solve for y. The result will only be a function if it passes the vertical line test.
Another way to see if an inverse of a function is a function is to look at your given function and determine if it is 1 to 1. That is: for function y = f(x), given a value of y is there only one value of x? (We already know that given a value of x there is only one value of y, else f(x) is not a function.) If so then the inverse of f(x) will be a function.