Answer
Please see below.
Work Step by Step
Looking at the graph of $f$, we find that the piecewise function $f$ is of the following form:$$\begin{cases}\frac{1}{2}, & -2 \le x \le -1 \\ \frac{1}{2}x+1, & -1 \le x\le 0 \\ 1, & 0\le x \le 1 \\ -x+2 &, 1 \le x \le 3 \end{cases}.$$So the function $g(x)= \frac{1}{f(x)}$ must be of the form$$\begin{cases}2, & -2 \le x \le -1 \\ \frac{2}{x+2}, & -1 \le x\le 0 \\ 1, & 0\le x \le 1 \\ -\frac{1}{x-2} &, 1 \le x \le 3 \end{cases}.$$Thus, the graph of $g(x)=\frac{1}{f(x)}$ can be easily sketched (Please note that the function $g(x)$ has a vertical asymptote at $x=2$).
