# Inverse functions and preimages

The notation $$f^{-1}$$ usually denotes the compositional inverse function of $$f$$, that is, a function $$g$$ such that, for all $$x$$ in the domain of $$f$$, $$g(f(x)) = x$$, and, for all $$x$$ in the domain of $$g$$, $$f(g(x))=x$$. This is a special case of the convention that $$f^n$$ denotes the $$n$$-fold composition of $$f$$ with itself, but, confusingly, it is also used in some contexts where $$f^n$$ normally denotes $$n$$-fold pointwise multiplication of $$f$$ with itself; see Powers of trigonometric functions. The inverse function does not exist unless $$f$$ is injective.

In some contexts, $$f^{-1}$$ denotes the preimage function from the power set of the codomain of $$f$$ to the power set of the domain of $$f$$, defined by $$f^{-1}(Y) = \{x : f(x) \in Y\}$$. This exists for all functions, whether or not they are injective.

In other contexts, like when $$f$$ is an element of a ring of functions on a space, $$f^{-1}$$ can denote the multiplicative inverse of $$f$$. This only exists if $$f(x)$$ is invertible for all $$x$$.