Answer
See below.
Work Step by Step
Step 1. To obtain the graph of $f(x)=1-2^{-x/3}$ from $y=2^{x}$, stretch the curve horizontally by a factor of 3, reflect across the y-axis, reflect across the x-axis, and then shift 1 unit up. See graph.
Step 2. We can determine the domain is $(-\infty,\infty)$; the range is $(-\infty,1)$; and the horizontal asymptote is $y=1$.