I wanted to prove the following statement: in case of a polynomial of an even degree, there exists some $y\in\mathbb{R}$ for which there is NO $x\in\mathbb{R}$ such that $f(x)=y$.
The proof I initially came up with was as following. I thought of splitting the proof into two cases: with positive and negative leading coefficient. Then, we can can prove that the polynomial with a positive leading coefficient attains a global minimum, and hence our statement follows for every $y<y_{\min}$. Similarly, we can prove that each polynomial with a negative leading coefficient attains a global maximum, and hence the statement follows for every $y>y_{\max}$.
However, is there an alternative (or perhaps a simpler) way to prove this statement?
A polynomial is continuous and so bounded on any bounded interval.
$$\lim_{x\rightarrow \pm \infty} f(x) = \sigma \cdot \infty$$
where $\sigma$ is the sign of the leading coefficient and so $f(x)$ is bounded above or below accordingly.