Scientific Note: Clarifying the Domain of x raised to the power of x (x^x)
Many sources mistakenly state that the domain of the function x^x is only the set of positive real numbers. However, this overlooks a critical fact: the function is also defined at all negative integers.
For example:
When x = -1, then x^x = (-1)^(-1) = -1
When x = -2, then x^x = (-2)^(-2) = 1/4
These values are real and valid. Therefore, the correct domain of x^x in real numbers includes:
All positive real numbers
All negative integers
Why this matters:
Misunderstanding the domain of x^x can lead to incorrect conclusions in limits, continuity, and calculus problems.
Bonus insight:
The function “disappears” from the real number line between negative integers because it enters the complex plane, and reappears only at those specific integer points.
Final note on signs:
In advanced understanding, a number’s sign is not just positive or negative, but a direction from the zero reference point. Imaginary numbers are simply numbers that move in other directions—not imaginary in existence, just in orientation.