inverse function

An inverse function is the undo button for another function. If one function does a job, its inverse runs the job backwards and returns you to where you started. Putting on socks then shoes has an inverse: take off shoes then socks. Whatever the forward function did, the inverse reverses, step by step.

We write the inverse of f as f^{-1} (read 'f inverse'; it is not a reciprocal). It is defined by undoing: f^{-1}(f(x)) = x for every x in the domain, and f(f^{-1}(y)) = y for every y in the range. For an inverse to exist as a function, f must be one-to-one — each output must come from only one input — otherwise the undo step would not know which input to return. On a graph, the inverse is the mirror image of f reflected across the line y = x, since reflecting swaps the roles of input and output.

A frequent slip is to read f^{-1}(x) as 1/f(x). It does not mean that — the -1 here marks 'inverse operation', not a power. When a function is not one-to-one we often restrict its domain to make it so: x^2 has no inverse over all real numbers because 2 and -2 share an output, but on x >= 0 its inverse is the genuine function sqrt(x).