surjection

Surjections as right invertible functions

The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can be undone by f). In other words, g is a right inverse of f if the compositionfog of g and f in that order is the identity function on the domain Y of g. The function g need not be a complete inverse of f because the composition in the other order, gof, may not be the identity function on the domain X of f. In other words, f can undo or "reverse" g, but cannot necessarily be reversed by it. For example, ${\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} ,(x,y)\mapsto x}$ has right inverse ${\displaystyle g:z\mapsto (z,0);}$ the composition in the other order gives ⁠${\displaystyle g(f((x,y)))=(x,0)}$⁠, which equals ⁠${\displaystyle (x,y)}$⁠ only if ⁠${\displaystyle y=0}$⁠.