Answer
please see graph (blue curve),
asymptote of $h$:$\quad\quad y=2$
domain of $h=(-\infty,\infty)$
range of $h=(2,\infty)$
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.
$h(x)=e^{x-1}+2=f(x-1)+2$
so the graph of $h(x)$ (blue) is obtained by
shifting the graph of f(x) (red) right by 1 units, (dashed green)
and then up by 2 units, (blue)
Reading the graph,
asymptote of $h$:$\quad\quad y=2$
domain of $h=(-\infty,\infty)$
range of $h=(2,\infty)$