Answer
(a) See graph. domain $(-\infty,\infty)$, range $(2,\infty)$, horizontal asymptote $y=2$.
(b) $g^{-1}(x)=log_3(x-2)$, domain $(2,\infty)$, range $(-\infty,\infty)$, vertical asymptote $x=2$.
(c) see graph.
Work Step by Step
(a) To obtain the graph of $g(x)=3^x+2$ from $y=3^x$, shift the curve 2 units up. See graph. We can find its domain $(-\infty,\infty)$, range $(2,\infty)$, horizontal asymptote $y=2$.
(b) We can find the inverse as $g^{-1}(x)=log_3(x-2)$ with domain $(2,\infty)$, range $(-\infty,\infty)$, vertical asymptote $x=2$.
(c) see graph.