# How to plot or generate function $\left(-\frac{1}{a},\left(-a\right)^a\right)$

I could plot $$\left(-\frac{1}{a},\left(-a\right)^a\right)$$ when $$a<0$$ that fuction is $$y=x^{\frac{1}{x}}$$ in Desmos and Geogebra , but they didn't plot when $$0

I want to know how to plot or generate function $$\left(-\frac{1}{a},\left(-a\right)^a\right)$$ when $$0

p.s.

I got the problem to define $$x^y$$ when $$x<0$$ and $$y$$ is not an integer

At first, I tried to generalize a function with these values :

$$\left(-\frac{1}{1},-1\right) \left(-\frac{1}{2},4\right) \left(-\frac{1}{3},-27\right) \left(-\frac{1}{4},256\right)...$$

Is there any similar or the same function?

• The problem is defining $x^y$ when $x \lt 0$ and $y$ is not an integer: for example what do you think $(-2)^\pi$ might be, even approximately? Is that a real number? Is it a unique answer? – Henry Jan 11 at 10:02
• The domain of $y = x^{\frac{1}{x}}$ when dealing with real numbers is $x > 0$. – KM101 Jan 11 at 10:03

The problem is that $$y = x^{1/x}$$ is not always real. Take for example $$a = 1/2$$ or equivalently $$x=-2$$, in this case
$$y = (-2)^{-1/2} = \frac{i}{\sqrt{2}}$$
That's why Desmos is having a hard time plotting your function for $$a > 0$$