The question is incomplete. Here is the complete question:
Which of the functions have a range of all real numbers greater than or equal to 1 or less than or equal to -1? check all that apply.
A. ![y=\sec x](https://tex.z-dn.net/?f=y%3D%5Csec%20x)
B. ![y= \tan x](https://tex.z-dn.net/?f=y%3D%20%5Ctan%20x)
C. ![y= \cot x](https://tex.z-dn.net/?f=y%3D%20%5Ccot%20x)
D. ![y= \csc x](https://tex.z-dn.net/?f=y%3D%20%5Ccsc%20x)
Answer:
A. ![y=\sec x](https://tex.z-dn.net/?f=y%3D%5Csec%20x)
D. ![y=\csc x](https://tex.z-dn.net/?f=y%3D%5Ccsc%20x)
Step-by-step explanation:
Given:
The range is greater than or equal to 1 or less than or equal to -1.
The given choices are:
Choice A: ![y=\sec x](https://tex.z-dn.net/?f=y%3D%5Csec%20x)
We know that, the ![\sec x=\frac{1}{\cos x}](https://tex.z-dn.net/?f=%5Csec%20x%3D%5Cfrac%7B1%7D%7B%5Ccos%20x%7D)
The range of
is from -1 to 1 given as [-1, 1]. So,
![|\cos x|\leq 1\\\textrm{Taking reciprocal, the inequality sign changes}\\\frac{1}{|\cos x|}\geq 1\\|\sec x|\geq 1](https://tex.z-dn.net/?f=%7C%5Ccos%20x%7C%5Cleq%201%5C%5C%5Ctextrm%7BTaking%20reciprocal%2C%20the%20inequality%20sign%20changes%7D%5C%5C%5Cfrac%7B1%7D%7B%7C%5Ccos%20x%7C%7D%5Cgeq%201%5C%5C%7C%5Csec%20x%7C%5Cgeq%201)
Therefore, on removing the absolute sign, we rewrite the secant function as:
![\sec x\leq -1\ or\ \sec x\geq 1\\](https://tex.z-dn.net/?f=%5Csec%20x%5Cleq%20-1%5C%20or%5C%20%5Csec%20x%5Cgeq%201%5C%5C)
Therefore, the range of
is all real numbers greater than or equal to 1 or less than or equal to-1.
Choice B: ![y= \tan x](https://tex.z-dn.net/?f=y%3D%20%5Ctan%20x)
We know that, the range of tangent function is all real numbers. So, choice B is incorrect.
Choice C: ![y= \cot x](https://tex.z-dn.net/?f=y%3D%20%5Ccot%20x)
We know that, the range of cotangent function is all real numbers. So, choice C is incorrect.
Choice D: ![y=\csc x](https://tex.z-dn.net/?f=y%3D%5Ccsc%20x)
We know that, the ![\csc x=\frac{1}{\sin x}](https://tex.z-dn.net/?f=%5Ccsc%20x%3D%5Cfrac%7B1%7D%7B%5Csin%20x%7D)
The range of
is from -1 to 1 given as [-1, 1]. So,
![|\sin x|\leq 1\\\textrm{Taking reciprocal, the inequality sign changes}\\\frac{1}{|\sin x|}\geq 1\\|\csc x|\geq 1](https://tex.z-dn.net/?f=%7C%5Csin%20x%7C%5Cleq%201%5C%5C%5Ctextrm%7BTaking%20reciprocal%2C%20the%20inequality%20sign%20changes%7D%5C%5C%5Cfrac%7B1%7D%7B%7C%5Csin%20x%7C%7D%5Cgeq%201%5C%5C%7C%5Ccsc%20x%7C%5Cgeq%201)
Therefore, on removing the absolute sign, we rewrite the cosecant function as:
![\csc x\leq -1\ or\ \csc x\geq 1\\](https://tex.z-dn.net/?f=%5Ccsc%20x%5Cleq%20-1%5C%20or%5C%20%5Ccsc%20x%5Cgeq%201%5C%5C)
Therefore, the range of
is all real numbers greater than or equal to 1 or less than or equal to-1.