If $f,g : V \to V$ are linear maps on a (*edit*: finite dimensional) vector space $V$ satisfying $f(x)=0$ for some nonzero $x$, does it follow that $f(g(y))=0$ for some nonzero $y$? Wh$y$?

This is part of a larger proof that I'm stuck on. I'd prefer hints over full solutions, since this is part of a homework problem.