Answer
(a) $(-\infty,0)$.
(b) See graph
(c) $(-\infty,\infty)$, $x=0$.
(d) $f^{-1}(x)=-\frac{10^{x}}{2}$.
(e) $(-\infty,\infty)$, $(-\infty,0)$.
(f) See graph.
Work Step by Step
(a) Use the domain requirement(s) for the function $-2x\gt0$, we have $x\lt0$ or $(-\infty,0)$.
(b) See graph for $f(x)=log(-2x)$
(c) We can determine the range as $(-\infty,\infty)$ and asymptote(s) as $x=0$.
(d) We can find $f^{-1}(x)=-\frac{10^{x}}{2}$. In short $x=log(-2y)\longrightarrow y=-\frac{10^{x}}{2}$
(e) We can find the domain of $f^{-1}(x)$ as $(-\infty,\infty)$ and range as $(-\infty,0)$.
(f) See graph.