Given some map $f : A \to B$, the distinction between the use of $\to$ and $\mapsto$ is usually clear: the former is to indicate the map is from $A$, the domain, to $B$, the range (or codomain), while the latter is to express the image of an element, say $a \mapsto b$ for $a \in A$ and $b \in B$.
However, is there a common usage of one or the other (or neither) when describing the image of a specific subset of $A$?
For example, let $A = B = \mathbb{Z}$ and $f : A \to B$ given by $f(x) = x^{2}$, with $S = \{1, 2, 3\}$ a subset of $A$. Then we can write $f(S) = \{1, 4, 9\}$ to express the image of $S$ under $f$, but would it be correct to write this with arrows as $$ S \to \{1, 4, 9\} \hspace{20pt}\text{or}\hspace{20pt} S \mapsto \{1, 4, 9\}? $$ My guess would be the former because some might read the latter as "the element $S$ maps to the set of three elements $\{1, 4, 9\}$", but I would like some clarification.
You would write $f(S) = \{1,4,9\}$.
There is no reason for one of these other notations, since they are not more succinct. Furthermore they are unclear.
Using $\mapsto$ would seem to imply that $S$ is an element of the domain and gets sent to $\{1,4,9\}$, which is not technically what you are wanting (you could be looking at the function induced by $f$ on the power set of $A$, but that is an awfully complicated bit of machinery for something so elementary).
Using $\to$ is also awkward; $S \to \{1,4,9\}$ lacks a bit of context, since the function $f$ is not appearing in this statement. Writing $f: S \to \{1,4,9\}$ would come across as if you are redefining your symbol $f$ to refer to a new function with domain $S$.
TLDR; don't use $\to$ or $\mapsto$ to denote the image of a set under $f$.