= Solution
{c}
\b[Exactly $|a|>1$ works.] Composition with the <hyperbolic tangent> is smooth for every $a$. If $|a|>1$, set $R=\operatorname{arctanh}(1/|a|)$. For $|x|\ge R$, $|a\tanh x|\ge1$, so the result vanishes; it is therefore a <test function> with <support of a function> $[-R,R]$.
If $|a|\le1$, every finite $x$ satisfies $|a\tanh x|<1$. The given positivity makes $\phi(|a|\tanh x)>0$ for every $x$, so its support is all of $\mathbb R$. In particular $|a|=1$ is excluded even though the function tends to zero at infinity: decay is not <compact support>.
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