Inverse function

The inverse of a mathematical function is a second function which "undoes" the operation of the first. Not every function has an inverse, and those that do are called invertible (or bijective).

The existence of an inverse function is important in mathematics often because the function and its inverse give "dictionaries" by which one can translate information about the domain to the range and back again.

Examples

Inverse trigonometric functions

The trigonometric functions $\sin x$, $\cos x$, $\tan x$, $\sec x$, $\csc x,$ and $\cot x$ are not invertible functions on their usual domains. However, by restricting the functions to appropriate smaller domains, they become invertible. There are multiple possible restricted domains for each function on which they become invertible, which causes ambiguity in the definition of the inverse trigonometric functions. This ambiguity often creates confusion for the newcomer to trigonometry.