Inverse function

Not to be confused with multiplicative inverse or additive inverse.

A function f and its inverse f⁻¹. Because f maps a to 3, the inverse f⁻¹ maps 3 back to a.

In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by $f^{-1}.$

For a function $f\colon X\to Y$, its inverse $f^{-1}\colon Y\to X$ admits an explicit description: it sends each element $y\in Y$ to the unique element $x\in X$ such that f(x) = y.

As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the output, then divides the result by 5. Therefore, the inverse of f is the function $f^{-1}\colon \mathbb {R} \to \mathbb {R}$ defined by $f^{-1}(y)={\frac {y+7}{5}}.$