Answer
please see graph (blue curve),
asymptote of $g$:$\quad\quad y=$0
domain of $g=(-\infty,\infty)$
range of $ g=(-\infty,\ 0)$
Work Step by Step
Graph $ f(x)=2^{x}\qquad$ (red, dashed)
by plotting the points from the table and connecting with a smooth curve.
$g(x)=-2^{x}=-f(x)$
so the graph of $h(x)$ (blue) is obtained by
reflecting shifting the graph of f(x) (red)
about the x-axis
Reading the graph,
asymptote of $g$:$\quad\quad y=$0
domain of $g=(-\infty,\infty)$
range of $ g=(-\infty,\ 0)$