Does a function $f$ exists such that: $f(n-k) \ne \Theta(f(n))$ for some constant $k\geq1$? I have encountered the following question in my homework assignment in Data Structures course:
&quot;Does a function $f$ exists such that: $f(n-k) \ne \Theta(f(n))$ for some constant $k\geq1$ ?&quot;
I think no such function $f$ exists, but I do not know how to prove it (or give a counter-example if one exists).
Counterexample: $f(n)=n!$ As $f(n)$ is $n$ times bigger  than $f(n-1)$, it is clear that $f(n-1) \neq \Theta(f(n))$.