Answer
a)$$R_f=(-\infty, -1] \cup [1, \infty )$$
b)$$R_g=(-\infty, -3] \cup [3, \infty )$$
The proper viewing window is shown in the graph below.
Work Step by Step
The red graph is the graph of the function $f(x)=\sec (3x+\frac{\pi }{2})$, and the blue graph is the graph of the function $g(x)=3\sec \pi (x+ \frac{1}{2})$.
Please note that a secant function $y=A\sec (Bx-C)$ has range $(- \infty, -|A|] \cup [|A|, \infty )$ and period $\frac{2\pi }{B}$.
So the function $f(x)=\sec (3x+\frac{\pi }{2})$ has range$$(-\infty, 1] \cup [1, \infty ),$$and the function $g(x)=3\sec \pi (x+\frac {1 }{2})$ has range$$(-\infty, -3] \cup [3, \infty ).$$
