Answer
(a) $f^{-1}(x)=-\frac{1}{3}(x-1)$.
(b) $f$: $(-\infty,\infty)$ and $(-\infty,\infty)$. $f^{-1}$: $(-\infty,\infty)$ and $(-\infty,\infty)$.
(c) See graph.
Work Step by Step
(a) $f(x)=1-3x\longrightarrow y=1-3x\longrightarrow x=1-3y\longrightarrow y=-\frac{1}{3}(x-1)\longrightarrow f^{-1}(x)=-\frac{1}{3}(x-1)$. Check $f(f^{-1})=x$ and $f^{-1}(f)=x$
(b) Domain and range of $f$: $(-\infty,\infty)$ and $(-\infty,\infty)$. Domain and range of $f^{-1}$: $(-\infty,\infty)$ and $(-\infty,\infty)$.
(c) See graph.