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    Find a rational function with the specified properties

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    Asymptotes for rational functions: f(x) = P(x)/Q(x). Given the formula, reconstruct a rational function based on the following pieces of information:
    1. The rational function has two vertical asymptotes and one oblique (slant) asymptote.
    2. One vertical asymptote is at x = - 5
    3. The second vertical asymptote, which crosses the positive segment of the x axis, is supposed to be selected by you.
    4. You are also supposed to select an oblique asymptote such that it is an increasing line (positive slope) with a y-intercept other than (0, 0).
    5. Graph your reconstructed rational function.

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    Solution Preview

    You can write down f(x) as a sum of functions f1(x), f2(x), f3(x), that each satisfies one of the requirements. As long as satisfying one requirement does not interfere with another requirement, this will work. So, let's take:

    f1(x) = 1/(x+5)

    Then f1(x) has a singularity at x = -5 which gives rise to the vertical asymptote there.

    If we take:

    f2(x) ...

    Solution Summary

    This solution provides a detailed explanation for each of the problems.