If a function $f(n)=\Theta(g(n))$, does it follow that $f(n/k)=\Theta(g(n))$ for a constant $k$?
Suppose $f(n) = 2^n$. Then $f(n / 2) = 2^{n / 2} = \sqrt{2^n}$ and setting e.g. $g(n) = 2^n$ as well shows $f(n) \in \Theta(g(n))$ but $f(n / 2) \notin \Theta(g(n))$, so this does not hold in general.