# How do you find the domain, range, and asymptote for y = 3 + 2 csc ( x/2 - pi/3 ) ?

Jul 23, 2018

Domain: $\uparrow$ asymptotic $\downarrow$ $x \ne \left(2 k \pi + \left(\frac{2}{3}\right) \pi\right)$,
$k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$.
Range: $y \notin \left(1 , 5\right)$

#### Explanation:

As $\csc \left(. .\right)$ value $\notin \left(- 1 , 1\right)$,

$y = 3 + 2 \csc \left(\frac{x}{2} - \frac{\pi}{3}\right) \notin \left(2 \left(- 1\right) + 3 , 2 \left(1\right) + 3\right) = \left(1 , 5\right)$

$\csc \left(\frac{x}{2} - \frac{\pi}{3}\right)$ determines the domain

$\frac{x}{2} - \frac{\pi}{3} \ne$ kpi, k = 0, +-1, +-2, +-3, ... #

$\Rightarrow x \ne \left(2 k \pi + \left(\frac{2}{3}\right) \pi\right)$

The period = period of $\sin \left(\frac{x}{2} - \frac{\pi}{3}\right) = \frac{2 \pi}{\frac{1}{2}} = 4 \pi$.

Asymptotes:$\downarrow x = \left(2 k \pi + \left(\frac{2}{3}\right) \pi\right) \uparrow , k = 0 , \pm 1 , \pm 2 , \pm 3 , . .$

See graph, depicting all these aspects.
graph{(1/2(y-3) sin (x/2-pi/3) -1)(y-1)(y-5)(x+4/3pi)(x-2/3pi)(x^2-4(pi)^2)=0[-10 10 -6 11] }