The range of a rational function depends on the horizontal or slant asyumptotes because the range is about the x-value so it will be involve with the y-axis. The domain and range are different and opposite. For example, The function of x is 2 / (x-3). Because the degree on the bottom is bigger, the horizontal asymptote will be y = 0. The function r is defined for all values of x other than 3; so the domain is {x│x≠3}. From the graph we see that the range is {y│y≠0}. A pure analytical way to find the range is to find inverse of f-1 or to find the horizontal asymptotes. Since f-1 (x) = 3+ 2/x and the domain of f-1 is {x│x≠0}, the range of f is {y│y≠0}. Or because the horizontal asymptote is 0, R = {0}.
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