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] }