Answer
(a) $f^{-1}(x)=\frac{3x+6}{x-2}, x\ne2$
(b) see graph
(c) $f(x)$ domain $(-\infty,3)U(3,\infty)$ and range $(-\infty,2)U(2,\infty)$, $f^{-1}(x)$ domain $(-\infty,2)U(2,\infty)$ and range $(-\infty,3)U(3,\infty)$.
Work Step by Step
(a) This function $f(x)=\frac{2x+6}{x-3}, x\ne3$ is one-to-one. Find the inverse as the following: $y=\frac{2x+6}{x-3}\longrightarrow x=\frac{3y+6}{y-2}\longrightarrow f^{-1}(x)=\frac{3x+6}{x-2}, x\ne2$
(b) see graph
(c) $f(x)$ domain $(-\infty,3)U(3,\infty)$ and range $(-\infty,2)U(2,\infty)$, $f^{-1}(x)$ domain $(-\infty,2)U(2,\infty)$ and range $(-\infty,3)U(3,\infty)$.
