= Solution
Let the nonconstant <polynomial> have degree $d$ and leading coefficient $c$. If $d$ is odd, its limits at $+\infty$ and $-\infty$ have opposite signs, so the <intermediate value theorem> shows that its image is $\mathbb R$. If $d$ is even and $c>0$, it tends to $+\infty$ at both ends. Outside a sufficiently large compact interval it exceeds $f(0)$, while on that interval the <extreme value theorem> gives a global minimum $a$. Continuity and the intermediate value theorem then give image $[a,\infty)$. For $c<0$, applying this argument to $-f$ gives $(-\infty,a]$. These are \b[exactly the three possible images].
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