Relation between the Schwarz and Test function spaces

Question

Consider the map $\varphi_a: B(0,a) \rightarrow \mathbb{R}^n$ defined as: \begin{align*} \varphi_a(x) = \frac{x}{a^2-|x|^2} \end{align*} where $|.|$ denotes the $l_2$ norm. Now if we take a Schwarz function $s \in \mathcal{S}(\mathbb{R}^n)$ we can define a new map $\Phi: \mathbb{R}^n \rightarrow \mathbb{R}$ such that: \begin{align*} \Phi = s\circ\varphi_a \end{align*} in $B(0,a)$, and extended to $0$ outside. My guess is that the object I am getting is a $C^{\infty}_c(\mathbb{R}^n)$ function. Is this true in general?(it works for maps like $e^{-x^2}$) Also, can every $C_c^{\infty}(\mathbb{R}^n)$ function be factored in this way?