If $f:\mathbb R\to\mathbb R$ is a polynomial of odd degree, then for every real $y$ there is a real $x$ such that $f(x)=y$

Question

Prove that if the function $f: \mathbb R \to \mathbb R$ is an odd degree polynomial, for every number $y ∈ \mathbb R$ there exists such a number $x ∈ \mathbb R$ that $f(x) = y$. Prove that this is not true for any even polynomial.

I have trouble with this proof. I don't know how to write it for any degree of polynomial. How can I correctly prove it?

Answer 1

Hints:

Let's assume for now that the leading coefficient of $f$ is positive (otherwise, we can just take the polynomial $-f$ and be in the same boat). Then, use the facts:

• $\lim_{x\to\infty} f(x) = \infty$ and $\lim_{x\to-\infty} f(x) = -\infty$
• $f$ is a continuous function

Using these two facts, try and prove:

1. There exists some $x\in\mathbb R$ such that $f(x) > y$
2. There exists some $x\in\mathbb R$ such that $f(x) < y$
3. There exists some $x\in\mathbb R$ such that $f(x) = y$

Try and prove these $3$ things (in order!) and tell us how far you got and where you are perhaps still stuck.