I am having some hard time to find an example of a function $f(n)$ that satisfies $f(\Theta(n)) = \Theta(f(n))$ but do not satisfies the polynomial-growth condition (the book's definition is at the end), I don't know if it is some weird piece-wise function. Can anyone shed some light? It's from an exercise (4.7-1) of CLRS-4ed by Thomas H. Cormen et al.
A function $f(n)$ defined on all sufficiently large positive reals satisfies the polynomial-growth condition if there exists a constant $\hat{n} \ge 0$ such that the following holds: for every constant $\phi \ge 1$, there e exists a constant $d > 1 $ (depending on $\phi$) such that $f(n)/d \le f(\psi n) \le df(n)$ for all $1 \le \psi \le \phi$ and $n \ge \hat{n}$.