The Collatz or the $3n+1$ conjecture is open.
Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$ conjecture?
What about the case of numbers of form $3^x$ (simplest non-trivial non-polynomial) where $x\in\mathbb Z$. Do these satisfy Collatz conjecture?