Prove any function can be written as a composition between an injective and a surjective function.

Question

Given an arbitrary function $f:A\rightarrow B$, write it as a composition between an injective and a surjective function, respectively.

Answer 1

This one is pretty straightforward. Intuitively, all we have to do is pick a function that maps set $A$ to the image of your initial function, and a function that "extends" the image of $f$ to set $B$, namely the identity function.

Here, the functions are named "$map$" and "$ext$", respectively and are defined as follows:

$map:A\rightarrow Imf\subseteq B, map(x)=f(x);$

$ext:Imf\rightarrow B,ext(x)=x.$

Since $map$'s codomain is the exact image of $f$, and since both functions are defined by the same formula, it follows that $map$ is surjective.

Since $ext$ is the identity function, it is evidently injective.

$f=ext\circ map \quad \square$