Addition of maps not always the addition of evaluations?

Question

Something that has always confused me is what we mean when we write something like $(f + g)(x)$. Is this just shorthand for $f(x) + g(x)$? Similarly, is $(f \circ g)(x)$ the same as $f(g(x))$? I suppose I just don't understand what it could mean to do $f+g$ or $f \circ g$ otherwise. This came up in the context of endomorphism rings, and it seemed like these two notions were treated differently.

Answer 1

An endomorphism ring ${\rm End}(G)$ of an abelian group $G$ is equipped with two operations, addition $(f,g)\mapsto f+g$ defined by $(f+g)(x) = f(x)+g(x)$ for each $x\in G$, and function composition $(f,g)\mapsto f\circ g$ defined by (in your notation) $(f\circ g)(x) = f(g(x))$ for each $x\in G$. With these operations, the endomorphism ring is a ring with zero element given by the zero mapping and the unit element given by the identity mapping.