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!
[Math] $x^x$ graph – what does it look like
exponentiationgraphing-functions
Related Solutions
The surface of equation $$z=\sin x+\sin y$$ has the shape of an "egg tray". It has maxima $z=2$ on a periodic grid, and this is why the level curves $z=1$ are regularly spaced approximate circles.
Now if we replace $x$ by $x^2$, we deform space horizontally so that $x^2$ increases faster and faster, giving a "compression effect".
By replacing $y$ with $y^2$, we get the effect on both axis.
Of course, the picture is symmetric by reflection, as the square function is even.
Remains to explain the "spikes" against the axis. If $y$ is small,
$$\sin x^2+\sin y^2=1\to y\approx\sqrt{1-\sin x^2}=\sqrt2\left|\cos\frac{x^2}2\right|$$
shows them.
Best Answer
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.
WolframAlpha may even draw some 3D graphs as you asked for: