No, a function is defined as a set. A function f : A -> B is defined as a subset of A x B (where the set A x B is the direct product of A and B) s.t. for each a in A there exists a unique b in B with (a, b) in f.
EDIT: Pretty much everything in math is derived from sets (hence Set Theory), although Category Theory messes things up a bit.
The set described by a function does
describe the function, but it does not
identify it! In set theory, it is not a goal to return to the original function, although it's almost always trivial to do so, because of the simple fact that distinct functions with equivalent input-output are distinct only in their definition, and not in their behavior.
Set theory is nothing more than a mental aid for mathematics; it isn't
required for deriving anything. Because it's based simpler principles, it's possible to bypass it. Doing so would be equivalent to reducing an integral to a summation, or a derivative to a limit.