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.

A function $g$ like you wish can be given, but it is not "interesting", as far as I'm concerned. Note that a nonzero linear function $\mathbf R \to \mathbf R$ is one-to-one, as $f$ is not one-to-one, $g$ has to be zero on the image of $f$. But on $(-\infty,0)$ you can take $g$ to be what you like. That is for example, define $g \colon \mathbf R \to \mathbf R$ by 
$$ g \colon x \mapsto \begin{cases} 0 & x \ge 0\\ \sin x & x < 0 \end{cases} 
$$ 
then $g \circ f = 0$ is linear.
