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