What is an easy way to find the range for a rational function?

Question

I sometimes have some trouble finding the range of a rational function. What is an easy or good way to do this?

Answer 1

Here are things to consider:

1) If the rational function has a linear term (or a term which can equal zero) in the denominator this will cause a vertical asymptote and in the neighbourhood of the asymptote the function will go to plus and/or minus infinity. Check the sign of the function on either side of each asymptote to determine which infinities.

2) If the degree of the numerator is greater than the degree of the denominator then the function will approach plus and/or minus infinity as $x$ approaches plus or minus infinity. Consider the limits to determine the sign.

3) If the degree of the numerator is equal to the degree of the denominator then the function will approach an asymptote equal to the ratio of the coefficients of the highest powers of the numerator and denominator. You will need to check if this value should be excluded or included in the range.

4) If the degree of the numerator is less than to the degree of the denominator then the function will asymptote to zero. As above check if zero should be in the range.

5) If the range is finite (e.g. cases 3 or 4) then calculus can determine maximums and minimums of the function which can be compared against each other (and the asymptotes) to determine the minimum and maximum of the range.