In class, we are learning exponential functions.
The following inverse exponential problem is bothering me: $y=x^{-\frac{1}{9}}$.
When graphed, I feel that it should look like it does on Desmos:
But other software is giving me different results. Look at, for example, Google:
and WolframAlpha:
and my trusty TI-84 calculator (which may only look incorrect because of its low pixel density):
There appears to be unreal (imaginary) answers involved, as is shown in the WolframAlpha screenshot. But, as we've learned in class, with any odd root there shouldn't be unreal answers, even for negatives (for example, $(-64)^{\frac{1}{2}} = 8i$, but $(-64)^{\frac{1}{3}} = -4$).
Can you please explain to me which one is right? (Or, if all are right in their own respectable ways?)
When you have something like $f(x) = x^{\frac{-1}{9}}$ you have many choices as to what to plot. For example, let us consider the argument $x = -1$. There are 9 complex numbers that are of the form $(a + bi)^{-9} = -1$.
You may think think all we have is $f(-1) = -1$, however, we also have many imaginary roots.
Soon enough you will learn Euler's formula and you will figure out these other roots.
$$e^{i \theta} = \cos \theta + i \sin \theta$$.
For example, another possible root is $e^{\frac{-i \pi}{9}}$
See what happens when you raise that to the $-9$th power.
As for the graphing software, it's really up to them what to graph. (you should notice that you have clicked on the complex valued plot for Wolfram)
Your example of $(-64)^\frac{1}{2}$ has two roots, them being $8i$ and $-8i$.
Additionally, $(-64)^\frac{1}{3}$ has 3 roots. We have $4e^{\frac{i \pi}{3}}, \>-4, \>4e^{\frac{-i \pi}{3}}$.