Is graph of $(-a)^x$ possible

Question

Is the graph of $(-a)^x$ possible? If yes, then what is it? If no, why not? This question came into my mind while studying transformations of graphs.

Here $a$ is any arbitrary constant.

Answer 1

For $a>0$, if your graph is in the real numbers (and we take the principal root where possible), then the graph will only have values at rational points with the denominator being odd.

At these points, it will evaluate to $-(a^x)$. And as these points are dense within the reals, such a graph will appear to be the same as $-(a^x)$.

If one allows complex numbers, and take $x$ to be real, then one must be more specific as to the argument of $-a$. If we take $-1=e^{\pi i}$, then we get the graph of $$a^xe^{\pi xi}=a^x[\cos(\pi x)+i\sin(\pi x)]$$

We can represent this as two graphs - the real and imaginary components - in which case the real part will be $a^x\cos(\pi x)$.