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 ${\displaystyle f^{-1}.}$

For a function ${\displaystyle f\colon X\to Y}$, its inverse ${\displaystyle f^{-1}\colon Y\to X}$ admits an explicit description: it sends each element ${\displaystyle y\in Y}$ to the unique element ${\displaystyle x\in X}$ 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 input, then divides the result by 5. Therefore, the inverse of f is the function ${\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} }$ defined by ${\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}$

Let f be a function whose domain is the set X, and whose codomain is the set Y. Then f is invertible if there exists a function g from Y to X such that ${\displaystyle g(f(x))=x}$ for all ${\displaystyle x\in X}$ and ${\displaystyle f(g(y))=y}$ for all ${\displaystyle y\in Y}$.

If f is invertible, then there is exactly one function g satisfying this property. The function g is called the inverse of f, and is usually denoted as f⁻¹, a notation introduced by John Frederick William Herschel in 1813.

The function f is invertible if and only if it is bijective. This is because the condition ${\displaystyle g(f(x))=x}$ for all ${\displaystyle x\in X}$ implies that f is injective, and the condition ${\displaystyle f(g(y))=y}$ for all ${\displaystyle y\in Y}$ implies that f is surjective.

The inverse function f⁻¹ to f can be explicitly described as the function

Recall that if f is an invertible function with domain X and codomain Y, then