Iterated function

Question

Let $$f(x)=x−\frac{1}{x}$$ Find the number of real solutions to $f(f(f(f(x))))=1$. Do I evaluate it completely, or is there some other way. After third composition it got nasty, so I left it.

Answer 1

Observe first that for any number $a\in {\mathbb R}$ the equation $$ f(x)=a $$ has two real solutions: $$ x_{1,2}=\frac{1}{2}(a\pm\sqrt{a^2+4}). $$ First we solve the equation $$ f(x)=1$$ and get two solutions $a_1, a_2$, then we solve two equations $$ f(x)=a_i$$ and get four solutions $b_1, \ldots, b_4$. We have to do four steps, in the end we have $16$ real solutions of the equation $$ f(f(f(f(x)))=1. $$

It is useful to see the graph of $f$: this function is a bijection from $(-\infty,0)$ to ${\mathbb R}$ and also from $(0,\infty)$ to ${\mathbb R}$. This shows that all above solutions are distinct and we really have $16$ different real numbers in the end.

EDIT. As Robert Lewis observed, one has to be careful with $f(\pm 1)=0$. However in our procedure this situation does not appear.