Name to use for codomain/range/target

There are many questions about how best to teach functions, for example why don't we teach codomains in HS and should we teach them at all. On Math.SX there are questions about the "right" name for this concept, such as this one and that one.

My question is different. Assuming that one is going to introduce functions as a special type of relation between two sets $A$ and $B$, with $A$ the domain (there aren't other words for that, are there?), what should we call $B$, from a pedagogical point of view?

There is no right answer here, but I do feel like people have strong opinions about this issue that are rooted in good pedagogical practice, which is after all why we're lurking on this site, right? So perhaps this question can serve as a good place to gather the 'range' of good pedagogical reasons for choice of terminology in this case. Discussions about how this is resolved (or if it's unnecessary) in non-Anglophone contexts are also welcome.

For what it's worth, because of the ambiguity in the meaning of the term my main feeling is, as a professor of mine once said, to leave the range to the cattle.

• Domain is also called "source" in some circumstances, I think (but not native speaker so I might be confusing with French usage). – Benoît Kloeckner Jan 16 '16 at 10:24

I think "codomain" is great. If $f: A \to B$, then I have only ever heard "the range of $f$" defined as the subset of the codomain $\{b \in B : \exists a \in A \textrm{ with } f(a) = b \}$. It is helpful to have names for these things, and I think these are pretty much standard in higher level mathematics.