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

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Answer 1 · 2018-07-23T06:26:33

Domain: $uarr$ asymptotic $darr$ $x ne ( 2kpi + (2/3)pi )$ ,
$k = 0, +-1,+-2, +-3, ...$ .
Range: $y notin ( 1, 5 )$

As $csc(..)$ value $notin ( -1, 1 )$ ,

$y = 3 + 2 csc (x/2 - pi/3 ) notin ( 2 (-1) + 3, 2 (1) + 3 ) = ( 1, 5 )$

$csc ( x/2 - pi/3 )$ determines the domain

$x/2 - pi/3 ne$ kpi, k = 0, +-1, +-2, +-3, ... #

$rArr x ne ( 2kpi + (2/3)pi )$

The period = period of $sin ( x/2 - pi/3 ) = (2pi)/(1/2) = 4pi$ .

Asymptotes: $darr x = ( 2kpi + (2/3)pi ) uarr, k = 0, +-1, +-2, +-3, ..$

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] }

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