Answer
(a) $(-\infty,\infty)$.
(b) See graph
(c) $(1,\infty)$, $y=1$.
(d) $ f^{-1}(x)=-log_3(x-1) $.
(e) $(1,\infty)$, $(-\infty,\infty)$.
(f) See graph.
Work Step by Step
(a) We can find the domain of $f$ as $(-\infty,\infty)$.
(b) See graph for $f(x)=1+3^{-x}$
(c) We can determine the range of $f$ as $(1,\infty)$, asymptote(s) as $y=1$.
(d) $f(x)=1+3^{-x} \longrightarrow y=1+3^{-x} \longrightarrow x=1+3^{-y} \longrightarrow y=-log_3(x-1) \longrightarrow f^{-1}(x)=-log_3(x-1) $.
(e) We can find the domain of $f^{-1}$ as $(1,\infty)$, range as $(-\infty,\infty)$.
(f) See graph.