A choice function f is usually defined on acollectionof subsets of some set S, not on a fixed subset A of S. Then for every A in the collection, f(A) is in A.

In your scenario, strictly speaking, f cannot be applied to the whole A because f is defined on elements of A. You are right the the f(A) probably should be interpreted as {f(a) | a in A}, so, yes, f(A) is probably not an element of A or of S. Sometimes this application is denoted by f[A] to distinguish it from the regular application of a function to an element of the domain.