Is the "composition" of two dense subsets of functions dense?

Question

Given $F \subseteq C_C(\mathbb{R}^d, \mathbb{R}^p)$, $F$ is dense in $C_C(\mathbb{R}^d, \mathbb{R}^p)$ in the supremum norm $\|\cdot\|_\infty$. Also given $G \subseteq C_C(\mathbb{R}^p, \mathbb{R}^s)$, $G$ is dense in $C_C(\mathbb{R}^p, \mathbb{R}^s)$ in $\|\cdot\|_\infty$. Is the set $G \circ F := \{g \circ f: g \in G, f \in F, g \circ f \in C_C(\mathbb{R}^d, \mathbb{R}^s)\}$ dense in $C_C(\mathbb{R}^d, \mathbb{R}^s)$? Note that $d, p, s\in \mathbb{N}$ are not necessarily equal. Any help would be much appreciated. Thanks!