How to graph y=x^x by using calculus | Summary and Q&A

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October 3, 2017
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How to graph y=x^x by using calculus

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.

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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.

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