Why Can't I graph $(-1)^x$?

Question

I tried to graph $(-1)^x$ but got nothing. Then I went on and tried $(-1)^{3.2}$ and got -1 as the answer, but when I tried $x=3.3$ instead I got an error.

I know that when x is a natural number that is even we should get $(-1)^x=1$ and that we should get $(-1)^x=-1$ if $x$ is odd, but how do the decimals affect the answer?

Answer 1

You need to plot the real and imaginary parts separately and you certainly need a more sophisticated graphing tool. The blue plot is the real. Note that $-1=e^{i\pi}$ so if $x$ is real then $(-1)^x=e^{ix\pi}=\cos(x\pi)+i\sin(x\pi)$ which is a point on the unit circle. For example $(-1)^{3.2}==-0.809017-0.587785i$.

Answer 2

Hint: For a general base $a > 0$ we have:

$$a^x = e^{(\ln a) x}$$

If $a=-1$, what is $\ln (-1)$?

Any attempt at an answer requires complex numbers, which your calculator probably does not support.

Answer 3

Because $x$ can assume all the real values, even rational numbers, that is resulting in a root: when the index of the root is even, your are asking for an imaginary number. On the contrary you could plot $(-1)^n$ with $n\in\mathbb{N}$, which is one of the most used sequences.