Inverse function

An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse.

Definition

A function $g$ is the inverse function of $f$ if $f(g(x))=x$ for each value of $x$ in the domain of $g$, and $g(f(x))=x$ for each value of $x$ in the domain of $f$. The function $g$ is denoted as $f$^-1 ("inverse of f").

Notation

Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, $f^-1(x)$ is not equal to 1/f(x).

Relevant observations

• If g is the inverse function of f, then f is the inverse function of g.
• The domain of f^-1 is the range of f and the range of f^-1 is the domain of f.
• A function may not have an inverse function, but if it has, the inverse function is unique.