Injective Function - Injections May Be Made Invertible

Injections May Be Made Invertible

In fact, to turn an injective function f : X → Y into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range J = f(X). That is, let g : X → J such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as incl_J,Y g, where incl_J,Y is the inclusion function from J into Y.