Composition of nonlinear maps Let: $$f: V \to W$$ and $$g: W \to U$$
where V, W, and U are some vector spaces.
Can you please give an example (an interesting one, if possible) where $$g \circ f$$ is a homomorphism
where f or g are not homomorphisms, if such a thing exists. (*Edit: I apologize for my impreciseness, but here I implicitly assumed, hence the "interesting", that g o f was not necessarily an isomorphism between V and U i.e. that f and g were not necessarily bijective so that the resulting would not be such a triviality as the identity function - it can be an isomorphism so long as it is nontrivial)
In other words, I proved that a composition of linear maps is linear, but am asking for a nontrivial counterexample of the converse.
In particular, I was somewhat interested in whether a function g can be constructed such that:
$$f:x \mapsto x^2$$
$$g(f(x+y))=g(f(x))+g(f(y))$$
where we take domain and codomain of f and g to be the reals.

It's not very exciting but in general if $f:V \rightarrow W$ is nonlinear but invertible then $f\circ f^{-1} = Id_V$ is linear.
