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Thread: The inverse of a function

  1. #1
    Member Jones's Avatar
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    The inverse of a function

    Hi,

    I would like some help with a problem.
    Show that F(x) = \frac{4x^3}{x^2+1} has an inverse and find

    f^{-1}(2)

    How do you invert fractions like this?

    //Jones
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  2. #2
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    skeeter's Avatar
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    Quote Originally Posted by Jones View Post
    Hi,

    I would like some help with a problem.
    Show that F(x) = \frac{4x^3}{x^2+1} has an inverse and find

    f^{-1}(2)

    How do you invert fractions like this?

    //Jones
    f(x) = \frac{4x^3}{x^2+1} ... domain is all reals.

    f'(x) = \frac{4x^2(x^2+3)}{(x^2+1)^2} > 0 for all x, telling us that f(x) is a strictly increasing function.

    since f(x) is strictly increasing, it is 1-1 over its domain, and as such has an inverse function.


    next, note that f(1) = 2

    that means that f^{-1}(2) = 1
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  3. #3
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    Note, that this problem did NOT ask you to actually find the inverse function- and skeeter did not. However, since you asked:

    Start by witing the function as
    f(x) = y= \frac{4x^3}{x^2+1} and swapping x and y:
    x= \frac{4y^3}{y^2+ 1}
    Now solve for y:
    x(y^2+ 1)= 4y^3
    4y^3- xy^2+ x= 0

    Solve that cubic equation for y. That, of course, is the hard part and probably why the problem did not require you to find the inverse!
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  4. #4
    Member Jones's Avatar
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    Oh dang, i thought you had to solve the darn thing
    Last edited by mr fantastic; Dec 30th 2009 at 07:49 PM. Reason: m --> r
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