Oops, sorry. It is not "inverse of a homomorphism" but "homomorphism of an inverse".
That is, if ab= e1, the identity in the first group (ring, field?), then you want to show that f(a)f(b)= e2. Well, the whole point of a homomorphism is that f(ab)= f(a)f(b). What does that tell you if ab= e1?