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Thread: The Inverse of a Exponential/Rational Equation

  1. #1
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    The Inverse of a Exponential/Rational Equation

    Find a pattern for $\displaystyle f^{-1}(x)$ if
    $\displaystyle f(x) = \frac {a^x +1}{a^x -1}$
    where $\displaystyle a>0, a \neq 1$.
    I have no idea what the question is asking by "a pattern". Does it mean the limitations of the inverse?

    I also tried to solve it by switching the "x" values with "y" values and working from there but I got stuck at $\displaystyle x(a^y - 1) = a^y +1$.
    Last edited by chrozer; Nov 30th 2008 at 01:42 PM.
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  2. #2
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    $\displaystyle y = \frac{a^x + 1}{a^x - 1}$

    $\displaystyle x = \frac{a^y + 1}{a^y - 1}$

    $\displaystyle xa^y - x = a^y + 1$

    $\displaystyle xa^y - a^y = x + 1$

    $\displaystyle a^y(x - 1) = x + 1$

    $\displaystyle a^y = \frac{x+1}{x-1}$

    $\displaystyle y = \log_a\left(\frac{x+1}{x-1}\right)$
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