the usual way to prove 2 sets are equal is to show that each one contains the other.

so in order to prove that A = B, you show that any a in A is also in B, and that any b in B is also in A.

so, with this problem, i'll help you get started.

suppose x is in f(A U A'). this means that x = f(u), for some u that is in A U A'.

u being in A U A', means that either u is in A, OR u is in A' (or maybe both).

to show that x is in f(A) U f(A'), you need to show that either x is in f(A) (that is x = f(a) for some a in A), OR x is in f(A') (that is, x = f(a') for some a' in A').

i recommend you do two cases:

1) u is in A

2) u is in A'

and show that either way...f(u) is in one of f(A) or f(A'). that will be half the proof, then you go "the other way" and start with y in f(A) U f(A').