Answer
(a) $(-\infty,\infty)$
(b) See graph.
(c) $(-2,\infty)$, $y=-2$.
(d) $ f^{-1}(x)=log_4(x+2)-1$
(e) $(-2,\infty)$, $(-\infty,\infty)$.
(f) See graph.
Work Step by Step
(a) Given $f(x)=4^{x+1}-2$, we can find its domain as $(-\infty,\infty)$
(b) See graph.
(c) We can determine the range of $f$ as $(-2,\infty)$, asymptote(s) as $y=-2$.
(d) Find the inverse $f(x)=4^{x+1}-2 \longrightarrow y=4^{x+1}-2 \longrightarrow x=4^{y+1}-2 \longrightarrow y=log_4(x+2)-1 \longrightarrow f^{-1}(x)=log_4(x+2)-1$
(e) We can find the domain of $f^{-1}$ as $(-2,\infty)$, range as $(-\infty,\infty)$.
(f) See graph.
