What is the Domain of the f(x)^g(x) Function?

[karkas]

Homework Statement

We have to find the domain range (hope I'm using this right) of some functions, one of them being h(x)=f(x)^g(x), say h(x)=(x^2-4)^(x). I've been looking around and couldn't find the domain range of the x^x function, so I am kinda stuck on this one.

Homework Equations

Find the domain of h(x)=(x^2 -4)

The Attempt at a Solution

Well there's a somewhat obvious requirement (I think), that x^2-4>0 . So we get x>2 or x<-2. Other than that though I don't see any other requirement that stems from the exponent. Maybe that isn't even the case , however. Any suggestions? Thanks!

 

[SammyS]

x^x does pose a problem.

When x is a negative number, x^q, where q is a rational number, is only defined if the numerator of q is even (and it is applied before applying the denominator), or if the denominator of q is odd.

x^r is undefined if r is irrational.

To my knowledge, the usual practice is to consider the domain of x^x to be (0, ∞). While you can make a case for including a subset of the negative rational numbers, to do so puts a lot of holes in the domain for x < 0 .