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