How do you find the range of a rational function?

Answer 1

If the degree of the polynomial is odd, the range is all real numbers - for example, y = x5.

If the degree is even, use derivatives to find maxima or minima. You learn about derivatives, maxima and minima in any basic calculus course. Example:

y = x4 - 3x3

Take the derivative:

y' = 4x3 - 9x2

Solve for zero:

4x3 - 9x2 = 0

This will give you two maxima or minima; in this case, check at which of these points the function has the smallest value. Because of the positive coefficient of the leading term, the function values go from this point all the way to plus infinity.