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Which of the following functions has a domain of all there real numbers?

a) #y = cotx#
b) #y = secx#
c) #y =sinx#
d) #y = tanx#

Answer 1

C. #y= sinx#

We need to look for asymptotes here. Whenever there are asymptotes, the domain will have restrictions.

#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.

#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.

#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.

#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!

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Answer 2

Adam Aguirre

The function that has a domain of all real numbers is the one that does not have any restrictions on the input values. In mathematical terms, this means the function is defined for all real numbers. Therefore, if you have a list of functions, you would need to examine each one to determine if any restrictions exist on the input values. The function that does not impose any restrictions and is defined for all real numbers would have a domain of all real numbers.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Which of the following functions has a domain of all there real numbers?

a) #y = cotx# b) #y = secx# c) #y =sinx# d) #y = tanx#

a) #y = cotx#
b) #y = secx#
c) #y =sinx#
d) #y = tanx#