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

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

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

How can we write $x$ as a function of $y$?

2. ## 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)

3. ## 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.

4. ## 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
$$e^x = \tfrac12\left( y + \sqrt{y^2+4} \right)$$