How to prove that $f(x)=x+\frac{1}{x}$ is not cyclic?

Question

Let $f(x)=x+\frac{1}{x}$ and define a cyclic function as one where $f(f(...f(x)...))=x$.

How do prove that $f(x)$ is not cyclic?

What I tried was to calculate the first composition:

$f(f(x))=x+\frac{1}{x}+\frac{1}{x+\frac{1}{x}}=\frac{x^4+3x^2+1}{x^3+x}$

Intuitively, I feel that this is clearly not going to simplify down to $x$, but how can I prove this beyond reasonable doubt?

Answer 1

Hint: for positive $x$ $$ f(x) > x $$

Answer 2

Note that $ f(\frac{1}{x})=f(x) $, then $f$ is not injective.

If $f^n(x)=x, \ n\geq2, $ for all $x$, then $f$ is injective, which is a contradiction.