Extensions for "Invisible" Parts of Functions

Question

My graphing calculator won't graph functions like $$f(x)=(-2)^x$$ which I assume is because it exists in $\mathbb R$ only sporadically for fractional $x$ and because it fluctuates so quickly from integer to integer. However, so that I can graph a function like this, I've devised a way to graph its points for integer values of $x$. If I factor out the $-1$, I get $$f(x)=(-1)^x2^x$$ and, for integer values of $x$, $(-1)^x$ has the same values as $\cos \pi x$, so I let $$f_\approx(x)=2^x\cos\pi x$$ And it works! I can see the real values of my original function for all integer values of $x$. I can use this same method for other functions, like $$g(x)=x^x$$ That sometimes do not exist and become sporadic because some values of $x$ result in negative bases with fractional exponents. My extension of this function is $$ g_\approx(x) = \left\{ \begin{array}{lr} x^x & : x \ge 0\\ (-x)^x \cos \pi x & : x \lt 0 \end{array} \right.\\ $$ My question is this: my extensions accurately represent the values of $f$ and $g$ for $x \in \mathbb Z$, but for the values of $f$ and $g$ that do not exist, do $f_\approx$ and $g_\approx$ mean anything? Do they have any relationship with the complex values generated by $f$ and $g$ for those values of $x$?