Given function $$f(x) = \frac{1+x}{x}$$ find the nested function: $$ \underbrace{f\circ f\circ f\circ ...\circ f}$$ the $f$ is there $n$-times.
I've found that $f\circ f = \frac{2x+1}{x+1}$ and that $f\circ f\circ f =\frac{3x+2}{2x+1} $ but there doesn't really seem to be a pattern and I'm finding it hard to find the domain and the equation for the nested function. (I can see the pattern for functions $f\circ f$ and $f\circ f\circ f$ but it isn't applicable to $f\circ f\circ f\circ f$ and so on...)
For the $n^{th}$ composition the composite function is: $$\frac{(F_{n+2})x+F_{n+1}}{(F_{n+1})x+F_{n}}$$ Where $F_k$ is the $k^{th}$ Finonacci term starting from $0$.
For a proof of the same you can rely on mathematical induction.
The domain is: $$dom(f)\backslash\{-\frac{F_k}{F_{k+1}}:1\leq k\leq{n}\}$$