What is the domain of $x^x$ when $ x<0$

Solution 1:

$x^x$ is well defined as a real function for $$(0,\infty) \cup \{ -\frac{m}{2n+1}| m, n \in {\mathbb N} \}$$

Solution 2:

Set $y = -x$.

Thus, we have:

$x^x = (-y)^{-y}$ when $y \ge 0$

Or equivalently,

$$x^x = \frac{1}{(-y)^{y}}$$

Which can be simplified to:

$$x^x = \frac{1}{(-1)^{y} y^y}$$

Therefore, the domain of $x^x$ consists of both reals and complex numbers depending on the value of $(-1)^y$ or to be more precise depending on the value of $y$.

Solution 3:

For negative values of $x$, when $x$ is not an integer, you run into surly problems involving complex numbers. These entail a study of the complex log function and its branches.

Clearly, $x\mapsto x^x$ makes sense for positive $x$. It also makes sense for negative integer values of $x$. It is not defined at $0$.