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