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    Graphing of Function With Vertical & Oblique Asymptotes

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    1. When preparing to graph the rational function
    y(x), this algebra is done.
    (a) Specify, with reasons, the (largest) domain
    of y(x). (You needn't repeat algebra)
    (b) Find where the graph crosses the y-axis.
    y(x) = x^4 - 2x^3 - 3x^2 - 3x â?" 1
    x^3 - 3x^2 + x - 3
    = x + 1 -(x - 1)(x + 2)
    (x - 3)(x^2 + 1)
    = x + 1 â?" 1/x â?" 3 â?" 1/x^2 + 1
    (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.

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    Solution Preview

    1.a)The domain includes all real numbers except x=3. This is the only value of x that results in the function becoming undefined. Sine the solution is x<3 And x>3, the largest sub-interval of the total domain is x<3 since it contains three more units that ...

    Solution Summary

    This solution contains an analysis of a function that has both vertical and oblique asymptotes. x-intercept and y-intercept values are calculated. There is an attached file which shows the graph of the function to illustrate the explanation of all function behaviors. This is an excellent study to help understand graphing or rational functions that contain both vertical and oblique asymptotes.