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An inverse function, also known as an inverse or anti-function, is a function that reverses the effect of another function. Given a function f that maps a set A to a set B, an inverse function f^-1 (read as "f inverse") maps B back to A, such that for every element x in A, applying f and then f^-1 returns the original element x. In mathematical notation, this is expressed as f^-1(f(x)) = x for all x in A.

Not all functions have an inverse. A function must be both injective (one-to-one) and surjective (onto) to have an inverse. Injective means that each element in the domain maps to a unique element in the codomain, and surjective means that every element in the codomain has a preimage in the domain. If a function is bijective (both injective and surjective), it has a unique inverse.

Inverse functions are fundamental in various areas of mathematics and its applications. They are used to solve equations, understand the behavior of functions, and in the study of group theory and other algebraic structures. The concept of an inverse function is also crucial in the field of computer science, particularly in the design of algorithms and data structures.