How do you find the inverse of f(x) = e^x - e^-xf(x)=exex?

1 Answer
Jan 30, 2016

Let y=f(x)y=f(x) and rearrange into a quadratic in e^xex to find:

f^(-1)(y) = ln((y+sqrt(y^2+4))/2)f1(y)=ln(y+y2+42)

Explanation:

Let y = e^x-e^-xy=exex

Then y(e^x) = (e^x)^2-1y(ex)=(ex)21

So (e^x)^2-y(e^x)-1 = 0(ex)2y(ex)1=0

Using the quadratic formula we find:

e^x = (y+-sqrt(y^2+4))/2ex=y±y2+42

One of these roots is negative, requiring xx to be non-Real.

So only the positive root is useful for e^xex as a Real valued function of Real values of xx:

x = ln((y+sqrt(y^2+4))/2)x=ln(y+y2+42)

That is:

f^(-1)(y) = ln((y+sqrt(y^2+4))/2)f1(y)=ln(y+y2+42)