Left inverse

Definition

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

Remarks

A function with a nonempty domain has a left inverse exactly when it is injective. See injective iff it has a left inverse. A left inverse is in general not unique. Its values on $B \setminus f(A)$ can be any elements of $A$. If $g$ is a left inverse of $f$, then $f$ is a right inverse of $g$.

Explanations

L1 human

Let $f \colon \mathbb{N} \to \mathbb{N}$ be $f(n) = n + 1$. The function $g$ with $g(n) = n - 1$ for $n \ge 1$ and $g(0) = 0$ is a left inverse: $g(f(n)) = n$. But $f(g(0)) = 1 \neq 0$, so $g$ is not an inverse on both sides.