i wonder if this will sound like a completely ignorant question, i hope not.. i'm learning this from some sort of self study course which is based on doing exercises and studying solutions, and it's really over my actual school level.

i think i can kind of know what to do for the front half part, but look at

(ii) When f(x)=0, f(x)=0 satisfies (1)

how is that so? $\displaystyle f(x) \cdot f(y)$ will become $\displaystyle 0 $, but then how do i know that $\displaystyle f(x+y)=0$?