Answer
please see graph,
asymptote of $g$:$\quad\quad y=0$
domain of $g=(-\infty,\infty)$
range of $g=(0,\infty)$
Work Step by Step
Graph $ f(x)=2^{x}\qquad$ (red, dashed)
by plotting the points from the table and connecting wtih a smooth curve.
$g(x)=2^{x+1}=f(x$+$1)$,
so the graph of g(x) (blue) is obtained by
shifting the graph of f(x) by 1 unit to the left.
Reading the graph,
asymptote of $g$:$\quad\quad y=0$
domain of $g=(-\infty,\infty)$
range of $g=(0,\infty)$