Here is the problem:
Let $P(x)$ be a polynomial of degree $n$. If $n$ is even, prove that $P(x)$ cannot have an inverse over $\Bbb R$.
I am not exactly sure how to proceed from here. But I did start off a little bit. Since $n$ is an even, then there is some value $k$ such that $$n = 2k$$ But from here, I am not exactly sure how to proceed.
Some help is appreciated!
Assuem w.l.o.g. that the leading coefficient is positive. Then $P(x) \to \infty$ as $ x\to \pm \infty$. But any injective continuous function is strictly monotonic. These two facts contradict each other.