Answer
Odd-degree: Domain=$(-\infty,\infty)$; Range=$(-\infty,\infty)$
Even-degree: If the leading coefficient is negative, Range=$(-\infty,y_0], y_0$ maximum; if the leading coefficient is positive, Range=$[y_0,\infty),y_0$ minimum
Work Step by Step
An odd-degree polynomial function can be written in the form:
$$f(x)=a_{2n+1}x^{2n+1}+a_{2n}x^{2n}+\cdots+a_1x+a_0.$$
The domain of $f$ is the set of all real numbers:
$$\text{Domain}=(-\infty,\infty).$$
The end behavior of an odd-degree shows that one end of the function's graph approaches $-\infty$ and the other $+\infty$, so the range of the function is the set of all real numbers:
$$\text{Range}=(-\infty,\infty).$$
An even-degree polynomial function can be written in the form:
$$f(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\cdots+a_1x+a_0.$$
The domain of $f$ is the set of all real numbers:
$$\text{Domain}=(-\infty,\infty).$$
The end behavior of an odd-degree shows that both ends of the function's graph approach $-\infty$ if $a_{2n}<0$ or $+\infty$ if $a_{2n}>0$.
If $a_{2n}<0$ the function has a maximum at a point $(x_0,y_0)$, so the range of the function is:
$$\text{Range}=(-\infty,y_0].$$
If $a_{2n}>0$ the function has a minimum at a point $(x_0,y_0)$, so the range of the function is:
$$\text{Range}=[y_0,\infty).$$
