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Out of curiosity, what does the $x^x$ graph look like?

I physically cannot picture it when $x < 0$. Does such a thing exist or do we only define the domain to be $x > 0$ where it's just a usual steep exponential?

Any help is appreciated!

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    $\begingroup$ It's not really useful to define $x^y$ when $x<0$ and $y$ is not an integer. So the function $x^x$ really shouldn't be defined when $x<0$. If you allow complex values, then you still ought to be dealing with a multi-valued function. $\endgroup$ – Thomas Andrews Jan 3 '17 at 21:36
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    $\begingroup$ Try inputting "x^x" into WolframAlpha. $\endgroup$ – Frank Vel Jan 3 '17 at 21:37
  • $\begingroup$ @FrankVel Everytime I input it, it only shows for $x > 0$ $\endgroup$ – TripleA Jan 3 '17 at 21:38
  • $\begingroup$ @TripleA I realized it would only work for positive integers on Google, but WolframAlpha should still give you a graph for negative values. $\endgroup$ – Frank Vel Jan 3 '17 at 22:01
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    $\begingroup$ Here is GrafEq's take on it. $\endgroup$ – J. M. isn't a mathematician Jan 6 '17 at 7:38
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The function $f(x)=x^x$ usually isn't defined for $x<0$. Notice, for example, that

$$f(-1/2)=(-1/2)^{-1/2}=\sqrt{-2}$$

and square roots of negative numbers generally isn't a good thing when you are graphing.

Just for your curiosity, the graph may be found on desmos and for convenience, it is also below:

enter image description here

For $x<0$, one may, if persistent, have complex numbers, and the graph is given by WolframAlpha. Below is a snippet:

enter image description here

For more interesting graphs, you could modify the input, like here.


WolframAlpha may even draw some 3D graphs as you asked for:

enter image description here

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  • $\begingroup$ Apparently it's a 3D diagram for $ x < 0$, which is fascinating yet confusing $\endgroup$ – TripleA Jan 3 '17 at 21:41
  • $\begingroup$ @TripleA 3D? Are you intending to have $x$ be complex numbers? As I'd rather not... $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 21:42
  • $\begingroup$ I know it isn't conventional but it bugs me too much for me not to know $\endgroup$ – TripleA Jan 3 '17 at 21:43
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    $\begingroup$ @TripleA Could try checking my last sentence. May help. $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 21:47
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    $\begingroup$ @ziggurism It's assuming some principal branch, which can indeed give a continuous complex-valued function. As noted by the graph, this does not give real values when $x$ is rational with odd denominator though. $\endgroup$ – Simply Beautiful Art Jan 3 '20 at 4:02
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In real analysis, $$ x^x:=e^{x\ln x},\quad x>0. $$ by definition. And goolge tells you it looks like this:

enter image description here

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  • $\begingroup$ Hahaha, Google? For graphing? Wow, I didn't even know this... $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 21:52
  • $\begingroup$ If one searches x^x in Google, then the graph would come out. $\endgroup$ – user9464 Jan 3 '17 at 21:54
  • $\begingroup$ Wow, that's easy. Is it for general graphs? Or... hm... $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 21:55
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It's not just a usual steep exponential; it has a little dip at the beginning. We don't usually define it for $x < 0$, or at least not for all $x$; $x^x$ is only real for $x < 0$ if $x$ is an integer or a fraction with odd denominator.

For graphing, I recommend WolframAlpha (www.wolframalpha.com). Just type "graph x^x" in the window. You'll notice that WolframAlpha does define it for negative values, but the result is a complex number.

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  • $\begingroup$ But I want to know what it looks like for $x < 0$ $\endgroup$ – TripleA Jan 3 '17 at 21:39
  • $\begingroup$ @TripleA I don't know what you're typing into WolframAlpha - it shows $x < 0$ for me. Try typing "graph x^x for x < 0". $\endgroup$ – Reese Jan 3 '17 at 21:41
  • $\begingroup$ @TripleA: It's not usually defined there unless you use complex numbers. Wolfram Alpha shows what it looks like if you use complex numbers. But there's ambiguity because the log of a complex number is not a well-defined function. Wolfram Alpha just uses the most common (I believe) convention for choosing a log of complex numbers. $\endgroup$ – The_Sympathizer Jan 3 '17 at 21:41

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