1. When preparing to graph the rational function y(x), this algebra is done.
y(x) = (x4 - 2x3 - 3x2 - 3x - 1) / (x3 - 3x2 + x - 3)
= x + 1 - {(x - 1)(x + 2) / (x - 3)(x2 + 1)}
= x + 1 - {1/(x - 3)} - {(1/x2 + 1)}
(a) Specify, with reasons, the (largest) domain of y(x). (You needn't repeat algebra)
(b) Find where the graph crosses the y-axis.
(c) Give all vertical asymptotes, and the graph's behaviour near such asymptotes.
(d) Specify the equation of the oblique asymptote to the graph, and the co-ordinates of all points, if any, where the graph crosses this asymptote.
(e) Specify the intervals of the x-axis, corresponding to values of y(x) which are greater than the corresponding point on the oblique asymptote.