Answer
Please see graph
(and details in the "work step by step" section.
Work Step by Step
$g(x)=3^{x}-1=f(x$)$-1$,
so the graph of g(x) (blue) is obtained by
shifting the graph of f(x) (red)
down by 1 unit.
Reading the graph,
asymptote of $ f:\quad y=0$
asymptote of $ g:\quad y=-1$
domain of $ f:\quad (-\infty,\ \infty)$
range of $ f:\quad (0,\infty)$
domain of $ g:\quad (-\infty,\ \infty)$
range of $ g:\quad (-1,\infty)$
