Math notation nomenclature

In high school algebra we learn the f(x) notation for functions. But in later courses there’s a more specific notation where we specify the sets being related, as in f: NZ. And this notation sometimes uses a “maps to” arrow for the definition that looks like this: |–>.

Is there a name for this notation that differentiates it from the f(x) notation?

Not that I’ve ever heard of, but that doesn’t mean it’s not out there.