Answer
please see graph (blue curve),
asymptote of $h$:$\quad\quad y=-1$
domain of $h=(-\infty,\infty)$
range of $h=(-1,\infty)$
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.
$h(x)=e^{x+1}-1=f(x+1)-1$
so the graph of $h(x)$ (blue) is obtained by
shifting the graph of f(x) (red)
left by 1 units, (dashed green),
and then down by 1 unit, (blue).
Reading the graph,
asymptote of $h$:$\quad\quad y=-1$
domain of $h=(-\infty,\infty)$
range of $h=(-1,\infty)$