Answer
For the function $csc^{-1}~2x$:
The domain is $(-\infty, -0.5]\cup[0.5,\infty)$
The range is $[-\frac{\pi}{2},0)\cup (0, -\frac{\pi}{2}]$
We can see a sketch of the graph of $~~csc^{-1}~2x~~$ below.
Note there is a horizontal asymptote at $y = 0$
Work Step by Step
Consider the function $csc~2x$:
The domain is all real numbers except $~~\frac{\pi}{2}~n~~$, where $n$ is an integer
The range is $(-\infty, -1]\cup[1,\infty)$
We can consider the function $~~csc^{-1}~2x~~$ with the domain restricted to $[-\frac{\pi}{4},0)\cup (0, -\frac{\pi}{4}]$
Then for the function $csc^{-1}~2x$:
The domain is $(-\infty, -0.5]\cup[0.5,\infty)$
The range is $[-\frac{\pi}{2},0)\cup (0, -\frac{\pi}{2}]$
We can see a sketch of the graph of $~~csc^{-1}~2x~~$ below.
Note there is a horizontal asymptote at $y = 0$
