A range-restriction of a function? Let $f:A \longrightarrow B$ be a function. What do we call the function (if it has a name !) $f:A \longrightarrow C$ where $C\subset B$. A "range-restriction of $f$"?
 A: It really depends on who is talking what the restricted range function would be. Many authors omit this altogether. For instance $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x)=\sin(x)$. Many authors will define it as above but many will just haphazardly define the co-domain to be whatever they want (on-the-fly). But formally $g:\mathbb{R} \to [-1,1]$ defined by $g(x)=\sin x$ is a defferent function than $f$. 
I don't think there is a single standard notation for this besides using the arrow between the domain and codomain.
A: It may not even be a function.  If there is some $a \in A$ that gets mapped to a point in $B \setminus C$ it isn't a function any more.  You can start with  the set of points that have image in $C$.  This is often denoted $f^{-1}(C)$ even though $f$ may not have an inverse.  You might call that $A'$ and then you can talk about $f$ restricted to $A'$, which we write $f_{\big|A'}$.
A: If you are working in a part of mathematics where things are phrased in categorical terms, you could call such a function a lift of $f$ by the inclusion map $C\rightarrow B$.
If you are not working in such a field, you're probably best off just saying "range restriction" and explaining what you mean. Possibly, you're working in a field where the codomain of a function isn't even that relevant, and you can just say that $f$ is already a function $A\rightarrow C$ if $f(a)\in C$ for all $a\in A$.

It's worth naming the opposite process: If you have a function $g:A\rightarrow C$, you can expand its range to $B$ by composing $g$ with the inclusion map $i:C\rightarrow B$ to get a function $i\circ g:A\rightarrow B$. You are trying to describe the opposite process.
That is, if you start with a function $f:A\rightarrow B$, you are trying to find a function $g:A\rightarrow C$ such that $f=i\circ g$. This is relationship can be described by saying that $g$ is a lift of $f$ through $i$ (or, if the existence of such a $g$ is all you care about, that $f$ factors through $i$).
That said, the word "lift" might give the wrong idea because it has pretty heavy connotations about either being a process to derive something interesting from a commutative diagram or being a process to "undo" some sort of quotient. If you're not in a field where these things are relevant (or where people talk about inclusion maps for other reasons), being more plainly spoken is advisable.
