definitions

Let $f: X \to Y$.

A function $g: Y\to X$ is said to be:

• A left inverse of $f$, if $g(f(x)) = x$, for all $x\in X$
• A right inverse of $f$, if $f(g(y)) = y$, for all $y \in Y$

relationship to injective and surjective functions

Let $f: X\to Y$. Then:

• $f$ has a left inverse if and only if $f$ is injective
• $f$ has a right inverse if and only if $f$ is surjective

proof that $f$ has a left inverse if it is injective

Let $f: X \to Y$ be injective. Let $x_0$ be any element of $X$. Define $g: Y\to X$ as such: $g(y) = \begin{cases}x \text{ if } y \in \text{range of }f \text{ and } y = f(x)\\ x_0 \text{ if }y \notin \text{range of }f\end{cases}$

$g$ is indeed a function from $Y$ to $X$. This is because if $y$ is in the range of $f$, then there is precisely one $x \in X$ with $y = f(x)$ because $f$ is injective.

Next, $g(f(x)) = x$ because $f(x)$ is in the range of $f$ and the first branch of the definition of $g$ applies to give $g(f(x)) = x$.

proof that $f$ is surjective if it has a right inverse

Let $f: X\to Y$.

Let $g: Y\to X$ be a right inverse of $f$. So $f(g(y)) = y$ for all $y \in Y$.

Given any $y \in Y$, we must produce an $x \in X$ with $f(x) = y$ to show $f$ is surjective.

If we choose $x = g(y)$, then $f(x) = y$ since $f(g(y)) = y$. So $f$ is surjective since we can produce an $x \in X$ for all $y \in Y$.