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Thread: $e^x - e^{-x}$

  1. #1
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    Question $e^x - e^{-x}$

    if
    $$y = e^x - e^{-x}$$

    How can we write $x$ as a function of $y$?
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  2. #2
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    Re: $e^x - e^{-x}$

    divide each side by 2

    y/2 = (e^x - e^-x)/2

    the right side is sinh x

    y/2 = sinh x

    take sinh x inverse

    x = sinh^-1(y/2)
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  3. #3
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    Re: $e^x - e^{-x}$

    Also
    \begin{align*}
    y &= e^x - e^{-x} \\
    y e^x &= (e^{x})^2 - 1 \\
    (e^{x})^2 - y e^x - 1 &= 0 \\
    e ^x &= \frac{y \pm \sqrt{y^2 + 4}}{2}
    \end{align*}
    And take the logarithm.
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  4. #4
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    Re: $e^x - e^{-x}$

    I should add that the solution with the negative root makes no sense in the real numbers (can you see why?). So we take
    $$e^x = \tfrac12\left( y + \sqrt{y^2+4} \right)$$
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