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Math Help - help me in this function question

  1. #1
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    help me in this function question

    if the function is
    f:[1,INFINITY]->[1,infinity] is defined by
    f(x)=2^(x(x-1)) then
    f inverse x=?
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  2. #2
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    y= 2^{x(x-1)}

    swap x and y and solve for y

    x= 2^{y(y-1)}

    y(y-1) = \log_2 (x)

    y^2-y = \log_2 (x)

    now complete the square

    \left(y^2-y +\frac{1}{4}\right) -\frac{1}{4}= \log_2 (x)

    \left(y-\frac{1}{2}\right)^2 -\frac{1}{4}= \log_2 (x)

    \left(y-\frac{1}{2}\right)^2 = \log_2 (x)+\frac{1}{4}

    y-\frac{1}{2} = \sqrt{\left(\log_2 (x)+\frac{1}{4}\right)}

    y = \sqrt{\left(\log_2 (x)+\frac{1}{4}\right)}+\frac{1}{2}

    therefore

    f^{-1}(x) = \sqrt{\left(\log_2 (x)+\frac{1}{4}\right)}+\frac{1}{2}
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