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^{2x}+1=f(2x)+$1
so the graph of $h(x)$ (blue) is obtained by
horizontally shrinking f(x) by factor $\displaystyle \frac{1}{2} ... $(green dashed)
then shifting up by $2$ units (blue).
Reading the graph,
asymptote of $h$:$\quad\quad y=2$
domain of $h=(-\infty,\infty)$
range of $h=(2,\infty)$