x^x = exp (x ln x)
we can say that ln is increasing and x is increasing and exp is increasing so x ln x is increasing and also exp(x ln x) is increasing by composition
but for the domain it is not all real numbers except zero
it is all positive real numbers except zero and for negative integers
for negative numbers it is defined for some rational n/p
eg (−1/3)^( −1/3) is not the same as (−2/6)^(−2/6).
so to avoid problem it is defined for all positive real numbers except zero