Domain of a function

The domain of the function:

(1) f(x) = x^x

is known to be [0 .. inf).
Considering that x = 2x/2, it is possible to rewrite the function as:

(2) f(x) = x^(2x/2)

Using the multiplicative property of the exponentials it is possible to rewrite it again as:

(3) f(x) = (x^2)^(x/2)

The function (3) is now defined over the whole real axis, as x^2 > 0, for all x.
The online plotting tools agree with this and draw the functions (1) and (3) on two different domains.
Since a function cannot have two domains, I suppose I wrote something wrong and the two functions (1) and (3) are indeed different.
Unfortunately, I can't find what's wrong... the mathematics seems easy...

Thanks for helping and regards,
Marco

 

there is no need to rewrite given function...domain of f(x)=x^x is all positive values of x:
{x element R : x>0} (all positive real numbers) (assuming a function from reals to reals) or interval (0,infinity)