Answer
(a) $(-2,\infty)$.
(b) See graph
(c) $(-\infty,\infty)$, $x=-2$.
(d) $f^{-1}(x)=3^{x-3}-2$.
(e) $(-\infty,\infty)$, $(-2,\infty)$.
(f) See graph.
Work Step by Step
(a) Use the domain requirement(s) for the function $x+2\gt0$, we have $x\gt-2$ or $(-2,\infty)$.
(b) See graph for $f(x)=3+log_3(x+2)$
(c) We can determine the range as $(-\infty,\infty)$ and asymptote(s) as $x=-2$.
(d) We can find $f^{-1}(x)=3^{x-3}-2$. In short $x=3+log_3(y+2)\longrightarrow y=3^{x-3}-2$
(e) We can find the domain of $f^{-1}(x)$ as $(-\infty,\infty)$ and range as $(-2,\infty)$.
(f) See graph.
