Question 115
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an arbitrary even function and $g: \mathbb{R} \rightarrow \mathbb{R}$ an arbitrary odd function. Define the function $\varphi: \mathbb{R} \rightarrow \mathbb{R}$ by the rule $\varphi(x)=f(g(x))+g(f(x))$ for all $x \in \mathbb{R}$. Which of the following equalities certainly holds for all real numbers $x$?
(A) $\varphi(x)=f(f(x))$
(B) $\varphi(x)=g(g(x))$
(C) $\varphi(x)=\varphi(-x)$
(D) $\varphi(-x)=-\varphi(x)$
(E) $\varphi(x)=2 f(g(x))$
(A) $\varphi(x)=f(f(x))$
(B) $\varphi(x)=g(g(x))$
(C) $\varphi(x)=\varphi(-x)$
(D) $\varphi(-x)=-\varphi(x)$
(E) $\varphi(x)=2 f(g(x))$