Problem: Given two natural numbers $x,y$ and two bijective linear functions $f,g$ on the same domain $\mathbb{N}$. Is there an arbitraty sequence of those composed functions, such that $y$ can be obtained from $x$?
Example: $f(x)=2x$, $g(x)=x+3$. Let $x=2, y =11$. Then $g\circ f\circ f (2)$ would be a answer. If you choose $y=12$ instead there is no answer.
Has this problem any specific name? Are there proof techniqes to show that there exists such a sequence or not (not necessary to give the concrete sequence)?
What if addtionaly $f$ and $g$ are commutative?