First edition was: Let $f(x)$ be a polynomial such that $f(x)$ and $f(x)-x$ have only one real root. How to prove, without derivatives, that 
$f(f(x))-x$ also has only one real root?

Second edition: Let $f(x)=a\_3 x^3 + a\_2 x^2 + a\_1 x + a\_0$. All $a\_i \neq 0$. $f(x)$ and $f(x)-x$ have only one real root. How to prove, without derivatives, that 
$f(f(x))-x$ also has only one real root?