#### Answer

For the function $~~2~cot^{-1}~x$:
The domain is $(-\infty, \infty)$
The range is $(0,2\pi)$
We can see a sketch of the graph of $~~2~cot^{-1}~x~~$ below.
Note there is a horizontal asymptote at $y = 0$ and $y = 2\pi$

#### 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 $(0,\pi)$
The general shape of the graph of $~~2~cot^{-1}~x~~$ is similar to the graph of $~~cot^{-1}~x~~$, but each y-value is doubled. Then for the function $~~2~cot^{-1}~x$:
The domain is $(-\infty, \infty)$
The range is $(0,2\pi)$
We can see a sketch of the graph of $~~2~cot^{-1}~x~~$ below.
Note there is a horizontal asymptote at $y = 0$ and $y = 2\pi$