Left/Right Inverses: Injective ⟺ Left Inverse, Surjective ⟺ Right Inverse

A function $f$ has a left inverse ($g\circ f=\mathrm{id}$) exactly when it is injective, and a right inverse ($f\circ g=\mathrm{id}$) exactly when it is surjective. So "is $f$ invertible on one side?" is just a disguised injectivity/surjectivity question.