Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.1 Inverse Circular Functions - 6.1 Exercises - Page 265: 74

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 $y = 2\pi$ 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 $y = 2\pi$ 