Let $f(x)$ be a quadratic function. Suppose $f(x)=x$ has no real root. Prove that $f(f(x))=x$ has no real root.
My attempt: Let the complex numbers $z$ and $\bar z$ be the roots of $f(x)=x$. I was able to conclude that $z$ and $\bar z$ are also the roots of $f(f(x))=x$.
But, I am unable to say anything about the other two roots of this quartic equation.
Please help me proceed.