Find the inverse of this function...y= ((e^x) - (e^-x))/2?

Answer 1

Well, this function is y=sinh(x) (that is hyperbolic sine)
The inverse of it is y=arcsinh(x).

I know that seems kind of trivial, but that's what it is. No BSing you.

I don't think you can do it in traditional methods.

EDIT:
I guess you can. I found this online:
arcsinh x = ln [x + (x^2 + 1)^(1/2)]

Viola!

Answer 2

the function is sinh x
If you put e^x=z
2y = z-1/z= so
z^2-2yz -1=0
as z must be positive(exponential)
z= (( 2y +sqrt(4y^2+4) = y+sqrt(1+y^2) so taking ln
x= ln( y +sqrt(y^2+1)
Interchange x and y
y=ln(x+sqrt(1+x^2) =Argshx