# How do you graph -2 tan^(-1) (x/4)?

Jan 4, 2017

Graphs are inserted and explained.

#### Explanation:

I have now adopted the convention ${\tan}^{- 1} \theta \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$ only.

$y = - 2 {\tan}^{- 1} \left(\frac{x}{4}\right) \in - 2 \left(- \frac{\pi}{2} , \frac{\pi}{2}\right) = - \left(- \pi , \pi\right)$,

giving 1 - 1 relation.

Inversely,

$x = - 4 \tan \left(\frac{y}{2}\right) , . y \in \left(- \pi . \pi\right) \mathmr{and} x \in \left(- \infty , \infty\right)$.

In the inserted graph, the required graph is restricted to

$y \in \left(- \pi . \pi\right)$.

This covers one y-period $\frac{\pi}{\frac{1}{2}} = 2 \pi$. The terminal asymptotes

are $y = \pm \pi$.

Graph of $y = - 2 {\tan}^{- 1} \left(\frac{x}{4}\right)$:

graph{x+4tan(y/2)=0[-500 500 -3.4 3.1416]}

Breaking the conventional 1 - 1 rule, 'one x - many y' graph is also presented. The terminal asymptotes are $y = \pm k \pi , k = 1 , 2 , 3. 4 , \ldots$. See below.

graph{x+4tan(y/2)=0[-500 500 -20 20]}