Every polynomial has a domain of (-∞,∞). The domain is the span of x values that are possible. This is easily proven by plugging in any x value. We always get some number for f(x).
The range is the extent of the possible 'y' values. For even functions (that is, the largest power of x is an even number) the range needs some farther analysis. For odd functions (the highest power of x is an odd number) the range is again (-∞,∞).
Think about the range for two functions you're probably already familiar with: f(x)=x and f(x)=x2
f(x) = x is a line, and the power of x is 1, which is an odd number. It is clear that the y values go from negative infinity to positive infinity. For f(x) = x2, the power is even. The graph of that looks like a smile with a clear bottom value for y, so it can't go to both positive and negative infinity.
I hope this helps!