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

## a) $y = \cot x$ b) $y = \sec x$ c) $y = \sin x$ d) $y = \tan x$

Jan 22, 2017

C. $y = \sin x$

#### Explanation:

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

A:

$y = \cot x$ can be written as $y = \cos \frac{x}{\sin} x$ by the quotient identity. There are vertical asymptotes whenever the denominator equals $0$, so if:

$\sin x = 0$

Then

$x = 0 , \pi$

These will be the asymptotes in 0 ≤ x < 2pi. Therefore, $y = \cot x$ is not defined in all the real numbers.

B:

$y = \sec x$ can be written as $y = \frac{1}{\cos} x$. Vertical asymptotes in 0 ≤ x < 2pi will be at:

$\cos x = 0$

$x = \frac{\pi}{2} , \frac{3 \pi}{2}$

Therefore, $y = \sec x$ does not have a domain of all the real numbers.

C:

$y = \sin x$

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:

$y = \tan x$ can be written as $y = \sin \frac{x}{\cos} x$, which will have asymptotes at $x = \frac{\pi}{2}$ and $x = \frac{3 \pi}{2}$ in 0 ≤ x <2pi#. It does not have a domain of all real numbers.

Hopefully this helps!