1 Answer

Nati Dykstein · Answer 1 · 2020-03-25T02:55:58+0000

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.
$y=\sec x$

B.
$y= \tan x$

C.
$y= \cot x$

D.
$y= \csc x$

Answer:

A.
$y=\sec x$

D.
$y=\csc x$

Explanation:

Given:

The range is greater than or equal to 1 or less than or equal to -1.

The given choices are:

Choice A:
$y=\sec x$

We know that, the
$\sec x=(1)/(\cos x)$

The range of
$\cos x$ is from -1 to 1 given as [-1, 1]. So,

$|\cos x|\leq 1\\\textrm{Taking reciprocal, the inequality sign changes}\\(1)/(|\cos x|)\geq 1\\|\sec x|\geq 1$

Therefore, on removing the absolute sign, we rewrite the secant function as:

$\sec x\leq -1\ or\ \sec x\geq 1\\$

Therefore, the range of
$y=\sec x$ is all real numbers greater than or equal to 1 or less than or equal to-1.

Choice B:
$y= \tan x$

We know that, the range of tangent function is all real numbers. So, choice B is incorrect.

Choice C:
$y= \cot x$

We know that, the range of cotangent function is all real numbers. So, choice C is incorrect.

Choice D:
$y=\csc x$

We know that, the
$\csc x=(1)/(\sin x)$

The range of
$\sin x$ is from -1 to 1 given as [-1, 1]. So,

$|\sin x|\leq 1\\\textrm{Taking reciprocal, the inequality sign changes}\\(1)/(|\sin x|)\geq 1\\|\csc x|\geq 1$

Therefore, on removing the absolute sign, we rewrite the cosecant function as:

$\csc x\leq -1\ or\ \csc x\geq 1\\$

Therefore, the range of
$y=\csc x$ is all real numbers greater than or equal to 1 or less than or equal to-1.

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