A range-restriction of a function?

Question

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 $f$"?

Answer 1

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.

Answer 2

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 $f_{\big|A'}$.

Answer 3

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 $a\in A$.

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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.