How to graph y=x^x by using calculus  Summary and Q&A
TL;DR
Graphing the function x to the x power requires considering the undefined nature of 0 to the 0 power, the change in base as well as exponent, and the need for x to be greater than zero to avoid complex numbers.
Questions & Answers
Q: Why is zero to the zero power undefined?
Zero to the zero power is undefined for regular computation. It does not provide a clear answer within the context of usual calculations.
Q: Can x be a negative number in x to the x power?
It is preferable to have x as a positive value in order to avoid complex numbers. Negative values for x result in the square root of a negative number, which is a complex solution.
Q: What is the domain of the function x to the x power?
The domain consists of x values that are greater than zero. Zero and negative numbers are excluded from the domain because they lead to undefined or complex solutions.
Q: Is the function always increasing?
The function appears to be increasing for x values greater than one. However, to determine if there are any maximum or minimum points, the derivative of the function needs to be analyzed.
Summary & Key Takeaways

Zero to the zeroth power is undefined, so the function is undefined at x=0.

The function is not an exponential function because both the base and the exponent change.

To avoid complex numbers, x must be greater than zero, as negative values result in imaginary solutions.

The domain of the function is x greater than zero.

The function has a relative minimum at x=1/e.