What is the domain and range of #y=csc x#?

1 Answer
Aug 30, 2017

Answer:

Domain: all #x in RR: x != n pi forall n inZZ#
Range: #(-oo, -1] uu [+1, +oo)#

Explanation:

#y = cscx#

#y = 1/sinx#

#sin x# is defined #forall x in RR#

#csc x# is defined wherever #sinx != 0 -> x != npi forall n in ZZ#

Hence, the domain of #y# is all #x in RR: x != n pi forall n inZZ#

Consider: #-1<=sinx<=+1 forall x in RR#

Hence, the local maxima of #y# are #1/-1 = -1#

and the local minima of #y# are #1/1 = 1#

Then, since #y# has no upper or lower bounds its range is:
#(-oo, -1] uu [+1, +oo)#

This maybe more easily visualised by the graph of #cscx# below.

graph{cscx [-8.89, 8.885, -4.444, 4.44]}