To determine which function has a domain of all real numbers, let's analyze each of the given functions:
A. [tex]\( y = \cot x \)[/tex] (cotangent):
- The cotangent function is defined as [tex]\( \cot x = \frac{\cos x}{\sin x} \)[/tex].
- This function is undefined when [tex]\( \sin x = 0 \)[/tex], which occurs at [tex]\( x = k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer.
- Thus, [tex]\( \cot x \)[/tex] is not defined for all real numbers because it has discontinuities at these points.
B. [tex]\( y = \sec x \)[/tex] (secant):
- The secant function is defined as [tex]\( \sec x = \frac{1}{\cos x} \)[/tex].
- This function is undefined when [tex]\( \cos x = 0 \)[/tex], which occurs at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer.
- Therefore, [tex]\( \sec x \)[/tex] is not defined for all real numbers due to these points.
C. [tex]\( y = \tan x \)[/tex] (tangent):
- The tangent function is defined as [tex]\( \tan x = \frac{\sin x}{\cos x} \)[/tex].
- This function is undefined when [tex]\( \cos x = 0 \)[/tex], which occurs at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer.
- Thus, [tex]\( \tan x \)[/tex] also has discontinuities and is not defined for all real numbers.
D. [tex]\( y = \sin x \)[/tex] (sine):
- The sine function is defined for all values of [tex]\( x \)[/tex] without any discontinuities.
- There are no points where [tex]\( \sin x \)[/tex] is undefined.
Given these points, the function that has a domain of all real numbers is [tex]\( y = \sin x \)[/tex].
Answer: D. [tex]\( y = \sin x \)[/tex]
