function composition - $n$ times

Question

Please consider this function:
$$f(x) = \frac{x}{{\sqrt[6]{{1 + {x^6}}}}} $$

What would be the value of the composition ($n$ times):
$$f \circ f\circ\cdots \circ f =\; ? $$

I tried doing it manually, maybe finding a pattern. I don't see another way solving it at the moment.

Answer 1

First, as it was mentioned in comments, observe what $f \circ f$ is $$ (f \circ f )(x)= \frac {\frac x{\sqrt[6]{1+x^6}}}{\sqrt[6]{1+\frac {x^6}{1+x^6}}} = \frac x{\sqrt[6]{1+x^6} \sqrt[6]{\frac {1 + 2x^6}{1+x^6}}} = \frac x{\sqrt[6]{1+2x^6}} $$ Now, assume that $$ f_k(x) = \underbrace{\left ( f \circ f \circ \cdots \circ f\right )}_{k\ \text{times}} (x) = \frac x{\sqrt[6]{1+k x^6}} $$ Try to find $f_{k+1}(x)$: $$ f_{k+1}(x) = (f \circ f_k) (x) = \frac {\frac x{\sqrt[6]{1+kx^6}}}{\sqrt[6]{1+\frac {x^6}{1+kx^6}}} = \frac x{\sqrt[6]{1+kx^6} \sqrt[6]{\frac{1+(k+1)x^6}{1+kx^6}}} = \frac x{\sqrt[6]{1+(k+1)x^6}} $$ Therefore, according to the principle of mathematical induction $$ f_n(x) = \underbrace{\left ( f \circ f \circ \cdots \circ f\right )}_{n\ \text{times}}(x) = \frac x{\sqrt[6]{1+n x^6}},\quad n \in \mathbb N $$

Answer 2

Hint: $$f(x)=\sqrt[6]{1-\frac1{1+x^6}}$$