Graph of $(-1)^x$

Question

What would the graph of $(-1)^{x}$ look like?

I know that the value of the function alternates between $1$ and $-1$ when it is defined so I think it would just be points spread over the lines $x=1$ and $x=-1$.

Is this correct? Also, will there be any definite pattern of the points.

Answer 1

First of all, it makes most sense to think of $(-1)^x$ as a complex number. One interpretation of this is to ask what the graph of $(-1)^z$ looks like in $\Bbb C$ as $z$ goes from $0$ to $1$. Since $-1=e^{i\pi}$, $(-1)^z=e^{i\pi z}$. As $z$ goes from $0$ to $1$, the value of the graph will travel along the unit circle from $1$ counter-clockwise to $-1$.

Answer 2

$(-1)^x$ isn't uniformly defined for values other than integers. For instance $(-1)^{(1/3)}=-1,$and$(-1)^{(1/3)}=.5+.5\sqrt3\cdot i$ But it's always true that $(-1)^x=\cos(\pi x)+i\sin(\pi x)$. Let me know when you want me to add a proof.

Answer 3

One possible definition is $$ (-1)^x=e^{\pi ix}=\cos(\pi x)+i\sin(\pi x) $$ However, we could be equally justified to say that $$ (-1)^x=e^{-\pi ix}=\cos(\pi x)-i\sin(\pi x) $$ or less commonly $$ (-1)^x=e^{3\pi ix}=\cos(3\pi x)+i\sin(3\pi x) $$ These all are based on the fact that $$ e^{(2k+1)\pi i}=-1 $$ for $k\in\mathbb{Z}$.

For $x\in\mathbb{R}$, all these definitions give results in the unit circle.