Question

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`

Explanation:

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!