Answer
For the function $arcsec~\frac{x}{2}$:
The domain is $(-\infty, -2]\cup[2,\infty)$
The range is $[0,\frac{\pi}{2})\cup (\frac{\pi}{2}, \pi]$
We can see a sketch of the graph of $~~arcsec~\frac{x}{2}~~$ below.
Note that $~~y = \frac{\pi}{2}~~$ is a horizontal asymptote.
Work Step by Step
Consider the function $sec~\frac{x}{2}$:
The domain is all real numbers except $\pi+2\pi~n$, where $n$ is an integer
The range is $(-\infty, -1]\cup[1,\infty)$
We can consider $~~sec~\frac{x}{2}~~$ with the domain restricted to $[0,\pi)\cup (\pi, 2\pi]$
Then for the function $arcsec~\frac{x}{2}$:
The domain is $(-\infty, -2]\cup[2,\infty)$
The range is $[0,\frac{\pi}{2})\cup (\frac{\pi}{2}, \pi]$
We can see a sketch of the graph of $~~arcsec~\frac{x}{2}~~$ below.
Note that $~~y = \frac{\pi}{2}~~$ is a horizontal asymptote.
