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

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.  
