Answer
(a) $(-\infty,\infty)$.
(b) see graph.
(c) $(0,\infty)$, $y=0$.
(d) $f^{-1}(x)=log_2(x)+3$.
(e) $(0,\infty)$ and $(-\infty,\infty)$.
(f) see graph.
Work Step by Step
Given $f(x)=2^{x-3}$, we have:
(a) the domain of $f(x)$: $(-\infty,\infty)$.
(b) see graph.
(c) From the graph, we can determine the range of $f(x)$: $(0,\infty)$, asymptote(s) of $f(x)$: $y=0$.
(d) $f(x)=2^{x-3} \Longrightarrow y=2^{x-3} \Longrightarrow x=2^{y-3} \Longrightarrow y=log_2(x)+3 \Longrightarrow f^{-1}(x)=log_2(x)+3$.
(e) we can find the domain and the range of $f^{-1}(x)$: $(0,\infty)$ and $(-\infty,\infty)$.
(f) see graph.