---
tag: 02486SD328
kind: definition
name: Left inverse
msc: ["03E20"]
defines: 05SG20AMXH
uses: [07HF9YCWPP, 0FSNQ0GNCC]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_function
    locator: Left inverses
---

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
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|0PVG1CRAJN]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].
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|0DDPZAG5ER]] 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.