We need to look for asymptotes here. Whenever there are asymptotes, the domain will have restrictions.
A:
#y= cotx# can be written as #y = cosx/sinx# by the quotient identity. There are vertical asymptotes whenever the denominator equals #0#, so if:
#sinx = 0#
Then
#x = 0, pi#
These will be the asymptotes in #0 ≤ x < 2pi#. Therefore, #y =cotx# is not defined in all the real numbers.
B:
#y = secx# can be written as #y = 1/cosx#. Vertical asymptotes in #0 ≤ x < 2pi# will be at:
#cosx =0#
#x = pi/2, (3pi)/2#
Therefore, #y = secx# does not have a domain of all the real numbers.
C:
#y = sinx#
This has a denominator of #1#, or will never have a vertical asymptote. It is also continuous, so this is the function we're looking for.
D:
#y = tanx# can be written as #y = sinx/cosx#, which will have asymptotes at #x = pi/2# and #x= (3pi)/2# in 0 ≤ x <2pi#. It does not have a domain of all real numbers.
Hopefully this helps!
