Right inverse

Definition

Let $f \colon A \to B$ be a function. A right inverse of $f$ is a function $h \colon B \to A$ such that $f \circ h = \mathrm{id}_B$, that is, $f(h(b)) = b$ for each $b \in B$.

Remarks

A function has a right inverse exactly when it is surjective. The proof uses the axiom of choice. See surjective iff it has a right inverse. A right inverse is in general not unique. A right inverse is injective.

Explanations

L1 human

For $f(x) = x^2$ from $\mathbb{R}$ to $[0, \infty)$, the square root $h(y) = \sqrt{y}$ is a right inverse: $(\sqrt{y})^2 = y$. The function $y \mapsto -\sqrt{y}$ is another right inverse.