$x^x$ graph - what does it look like?

Question

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!

Answer 1

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.

Answer 2

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:

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

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

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

Answer 3

In real analysis, $$ x^x:=e^{x\ln x},\quad x>0. $$ by definition. And goolge tells you it looks like this: