Which function has a domain of all real numbers? A.y=cotx B.y=secx C. y=sinx D.y=tanx

C. ##y= sinx##

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!