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Question:

akter
akter
Student
High School - 12th Grade

f(x)=logx & g(x)=x,so i need graph of the function f(x)/g(x)

Please describe these key features: intercepts, asymptotes, domain, range, symmetry, maxima, minima, slope.

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Posted by akter on Wednesday January 14, 2009 at 11:52 AM and tagged with asymptotes, domain, intercepts, math, maxima, minima, range, slope, symmetry.

Answers:

  1. mcetner
    mcetner Teacher
    High School - 12th Grade

    eNotes Editor

    You need to know the parts of the graph y = (logx)/x.

    If you are allowed to use your calculator, you can find intercepts, maxima, minima, all on there.  If not:

    Intercepts:

    • y intercept is when x = 0, but since this is log, there is an asymptote at x = 0, so no y-intercept.
    • x-intercept is when y = 0, so if 0=logx/x, 0=logx, and x = 1 (x-intercept is 1)

    Asymptotes: vertical asymptote at x = 0 (y-axis)

    Domain: x>0 (domain of logx)

    Range: all real numbers (range of logx)

    Symmetry: none, the function does not cross the y-axis, so it can't have y-axis or origin symmetry

    Maxima and Minima: the only way I know to do this without the calculator is by calculus. Take the derivative and you get (1-logx)/x^2, and relative extrema are where 1-logx = 0, so logx = 1, so x = 10.  This is a maximum because the function is increasing until there, decreasing after that point.

    Slope: increasing until x = 10, then decreasing.

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    Posted by mcetner on Saturday February 7, 2009 at 5:18 AM