# Inverse Function problem involving e^x

1. Sep 19, 2012

### ChromoZoneX

1. The problem statement, all variables and given/known data

Let g(x) = (e^x - e^-x)/2. Find g^-1(x) and show (by manual computation) that g(g^-1(x)) = x.

2. Relevant equations
g(x) = (e^x - e^-x)/2

3. The attempt at a solution

I get the inverse = ln[ (2x + sqrt(4x^2 + 4) ) / 2 ]

How do I proceed?

2. Sep 19, 2012

### Mentallic

First of all,

$$\frac{2x\pm \sqrt{4x^2+4}}{2}$$ can be simplified. Factor out 4 from the square root and cancel the 2's.

If g(x) = x then what is g(2x)? Then what is g(f(x))?

3. Sep 28, 2012

### ChromoZoneX

Thank you very much. I understood the question and answered it

PS: Sorry for the late reply.

4. Sep 28, 2012

### Ray Vickson

First: please use brackets, so write e^(-x) instead of e^-x and g^(-1) instead of g^-1 (however, e^x is OK as written). You want to find what x gives you g(x) = y; that would be g^(-1)(y). Just put z = e^x, so you have the equation (1/2)(z + 1/z) = y, which is solvable for z. After that, x = log(z).

BTW: to guard against confusing yourself and others, I suggest you refer to the argument of the inverse function as y (or z, or w or anything different from x), at least until you have obtained the final result. Then you can switch to any symbol you want. However, if your teacher wants you to do it another way, follow the requirements you are given.

RGV