Finding the inverse of a function?

Question

what is the inverse of the square root of 6 minus e^(2x)?
(6-e^(2x))^.5 if you're confused of how to format that. This was my incorrect course of action:

1. replace x with y. x = sq. root (6-e^(2y)
2. square each side. x^2 = 6-e^(2y)
3. subtract 6 from each side. x^2-6 = -e^(2y)
4. multiply each side by -1. 6-x^2= e^(2y)
5. mutiply each side by the natural logarithm. ln (6-x^2) = ln e^2y
6. Isolate y. [ln(6-x^2)]/2=y

That made sense to me but doesn't work when you graph it.

Oh, what is the domain of the inverse?

anyone else want to take a stab at it?

Answer 1

looks exactly right to me.
plug in x = sqrt(6-e^2y)
y = ln(6-(sqrt(6-e^2y)^2)/2 = ln(e^2y)/2 = 2y/2 = y

for the domain
x^2 < 6 to keep ln function real. -sqrt(6) < x < sqrt(6)

Answer 2

y=sqrt(6-e^(2x))
The domain of x is (- infinity, .5ln(6)]
Therefore this is also the range of the inverse, if there is an inverse. But the inverse you computed cannot have this range. Hence the function has no inverse.

The inverse of a function has a domain = range and a range = domain of the original function.