Question

inverse of (e^x - e^-x)/2

Answer 1

Set u = e^x and create a quadratic in u. Solve and you'll find the function (arcsinh) you want.

Answer 2

put
\[y=\frac{e^x+e^{-x}}{2}\] and solve for x via
\[2y=e^x-\frac{1}{e^x}=\frac{e^{2x}-1}{e^x}\]
\[2ye^x=e^{2x}-1\]
\[e^{2x}-2ye^x-1=0\] and solve this quadratic equation in
\[e^x\] as jamesj wrote

Answer 3

I.e., if y = (1/2)(e^x - e^-x) then y = (1/2)(u - 1/u)

Rearrange to obtain an equation in u. Then substitute back in e^x.

Answer 4

Right, crossed messages ... what sat73 just wrote as well.