Inverse function

In mathematics, an inverse function is a function that "reverses" another function. That is, if f is a function mapping x to y, then the inverse function of f maps y back to x.

Definitions

Let f be a function whose domain is the set X, and whose image (range) is the set Y. Then f is invertible if there exists a function g with domain Y and image X, with the property:

If f is invertible, the function g is unique, which means that there is exactly one function g satisfying this property (no more, no less). That function g is then called the inverse of f, and is usually denoted as f⁻¹.

Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse relation is the inverse function.

Not all functions have an inverse. For this rule to be applicable, each element y ∈ Y must correspond to no more than one x ∈ X; a function f with this property is called one-to-one or an injection. If f and f⁻¹ are functions on X and Y respectively, then both are bijections. The inverse of an injection that is not a bijection is a partial function, that means for some y ∈ Y it is undefined.