Show that if $f(2n) \le Cf(n)$ for large enough $n$, where $f:N \to N$. Then $f(n)$ is asymtotically bounded by a polynomial?

Question

Show that if $f(2n) \le Cf(n)$ for large enough $n$, where $f:N \to N$. Then $f(n)$ is bounded by a polynomial?

Some suggestions? I tried using the Master theorem but I didn't get some convincent.

Answer 1

Try $$ f(n) = \cases{n! & if $n$ is odd\cr 1 & if $n$ is even} $$