Codomain

Vs: image: the codomain is the set that the function might reach.

The image is the exact set that it actually reaches.

E.g. the function: could have:

Note that the definition of the codomain is somewhat arbitrary, e.g. $x^2$ could as well technically have codomain: even though it will obviously never reach any value in ${\mathbb{R}}^2$.

The exact image is in general therefore harder to characterize.

A function where the domain is the same as the codomain.