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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 $0<a$

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

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?

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  • $\begingroup$ 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? $\endgroup$ – Henry Jan 11 at 10:02
  • $\begingroup$ The domain of $y = x^{\frac{1}{x}}$ when dealing with real numbers is $x > 0$. $\endgroup$ – KM101 Jan 11 at 10:03
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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 $a > 0$

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