How do you graph a negative number raised to $x$? As a freshman in High School, this may be a ridiculous or simple question, but what does a graph look like of something like $y=(-1)^x$? While I imagine that the graph is more or less a dotted line across $y=1$ and $y=-1$, are there are undefined points? How do fractional exponents interact with their negative subjects?
 A: We normally can only graph functions that are continuous, or at least close to it (perhaps with a handful of points where they are not continuous).  You will study this property in calculus.  In brief, it means that if you vary $x$ only a little, then you will vary the value $f(x)$ only a little.
The function $(-2)^x$ is not continuous, unlike the function  $2^x$.
A: This question is indeed tricky.
starting with the natural and integer numbers is quite easy, there you get your pattern with dots on $\{-1,1\}$.
continuing to the rationals we realize we have a problem, as $\sqrt {-1}=(-1)^{\frac12}$ is not part of the reals. However some of them are, e.g. $(-1)^3 = -1$. So you get some additional points (in fact many) wher your funciton is defined. 
On all the irrational number your function is not defined, as we have the definition $a^b = e^{b\cdot \log(a)}$. Where $\log(-1)$ is not defined.
A: As others mention, this graph will have many points where it will not be defined. In any finite interval like $(0,1)$ there would be infinitely many points where this function is discontinuous.
Consider $f(x) = (-1)^{x}$  and $x = \frac{p}{q},$ where $p,q$ are mutually prime integers. Now if $q$ is even, $f(x)$ will not be real. 
