i know sinh(x) = (e^x - e^-x)/2
but what is sinhInverse(x) equal to?
First of all, the terminology is "hyperbolic" not "hyperbole."
We have a function
$\displaystyle y = \frac{e^x - e^{-x}}{2}$
To find the inverse we need to switch the roles of x and y:
$\displaystyle x = \frac{e^y - e^{-y}}{2}$
and solve for y:
$\displaystyle 2x = e^y - e^{-y}$ <-- Multiply through by $\displaystyle e^y$
$\displaystyle 2xe^y = e^{2y} - 1$
$\displaystyle e^{2y} - 2xe^y - 1 = 0$
This is a quadratic in $\displaystyle e^y$, so
$\displaystyle e^y = \frac{2x \pm \sqrt{4x^2 + 4}}{2}$
$\displaystyle e^y = x \pm \sqrt{x^2 + 1}$
$\displaystyle y = ln \left ( x \pm \sqrt{x^2 + 1} \right )$
We discard the "-" solution since the arguement of ln cannot be negative. ($\displaystyle x - \sqrt{x^2 + 1}$ is negative everywhere.)
So finally
$\displaystyle y = ln \left ( x + \sqrt{x^2 + 1} \right )$
This is the inverse function to sinh(x). (The dotted line in the graph below is the line y = x.)
-Dan