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