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Go ahead and load the desmos calculator: https://www.desmos.com/calculator

Now graph $x^x$

You should see something like this. An apparent continuation of the curve is shown.

ghost

Upon zooming in, it vanishes! What is causing Desmos to be haunted by ghosts?

vallev
  • 81

2 Answers2

1

The root issue is that the notation $x^x$ does not represent a well-defined real-valued function when $x\leq 0$.

Probably the most sensible continuation to negative numbers is via restriction of the complex function $F(x) = \exp(x \log x)$ to the real line. Of course $\log(x)$ is multi-valued so you could take e.g. the principal value. But this produces non-real results for some rational values of $x$ where one might "expect" a real result, i.e. $F(-1/3)$ is complex and does not equal $-\sqrt[3]{3}.$

What Desmos seems to be doing instead is extending $x^x$ to a real-valued function on the negative rationals with odd denominator (and leaving $f$ undefined at other negative $x$). This explains the sign alternation as well as why the function disappears when you zoom in: if Desmos happens to only sample $f$ at negative points that aren't rational with odd denominator, it won't draw anything because all sampled values are undefined.

user7530
  • 49,280
0

See this video which gives an explanation. The reason there is that for negative $x$ such that $x \geq -1$ and $\frac{1}{x}$ is even $x^x$ is imaginary.

stochs
  • 173