Left inverse ($g\circ f={\mathrm{id}}_X$)

A left inverse of a function $f:X\to Y$ is a function $g:Y\to X$ satisfyingIt forces injectivity: if $f(x)=f(x^{\prime })$, applying $g$ gives $x=x^{\prime }$. For example, $f(n)=2n$ on the natural numbers has left inverse $g(m)=\lfloor m/2\rfloor$. The reverse function composition is not the identity at odd inputs.

Conversely, an injective function with nonempty $X$ has a left inverse. On the image of a function $f(X)$, send each value to its unique preimage; outside the image, send every point to one fixed $x_0\in X$. The nonempty-domain qualification matters: the unique empty function from $\varnothing$ to a nonempty $Y$ is injective, but no function $Y\to \varnothing$ exists.