Inverse function

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 ${\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}$