Let $P(x)$ be an odd degree ploynomial in $x$ with real coefficients. Show that the equation $P(P(x))=0$ hasat least as may distict real roots as $P(x)=0$.
Proof
Assume that ${x_1},x_{2},...,x_{n}$ are the real roots of $P(x)=0$. So the roots of $P(P(x))=0$ are given by $$P(x)=x_i \space \forall i\in\{1,\ldots,n\}$$ Now as $P(x)$ is odd we can conclude that $P(x)$ maps from $\mathbb R \to \mathbb R $. So $P(x)=a$ has atleast one real root for all $a \in \mathbb R$.And therefore we can coclude that $P(P(x))=0$ must have atleast $n$ real distict roots.
Now my question is if we can extend this to even fucntions. The above method doesnt quite work now as even function might not map to $\mathbb R$.