There seems to be a name for a function with a specific codomain. Namely, if the codomain is $D$, then we called it a $D$-valued function. For example, we have complex valued functions, real valued functions, etc.
Is there an analogous name for functions with a particular domain? It is very cumbersome to say 'a function whose domain is...' every single time.
Usually in a written text the shortest and most common way would be to write out both domain and codomain, as in "let $f:A\to B$". I can't really think of a context where you care about the domain but not the codomain.
However, when the domain is $\Bbb R$ or $\Bbb C$ you have the common phrasing "function of a real variable", or "function of a complex variable". It is also used with multiple variables, as in "function of $n$ complex variables", etc.