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Chapter 4: Problem 95

Write the equation for a cosecant function satisfying the given conditions. $$\text { period: } 3 \pi ; \text { range: }(-\infty,-2] \cup[2, \infty)$$

Short Answer

Expert verified
The equation of the cosecant function satisfying the given conditions is \(y = 2csc((2/3)x)\).

Step by step solution

01

Determine basic function

The basic cosecant function is \(y = csc(x)\). The pattern of this function is \(y = 1/sin(x)\). We will use this basic function as a start and then apply the necessary transformations.
02

Transformation for period

The general form for a function cos or sin with a period of \(p\) is \(y = csc(Bx)\), where \(B = 2\pi/p\). We are given the period is \(3\pi\), so \(B = 2\pi/(3\pi) = 2/3\), and our function becomes \(y = csc((2/3)x)\). This changes the period of the basic function to \(3\pi\).
03

Transformation for the range

The range for the basic cosecant function is \((-鈭�, -1] \cup [1, 鈭�)\), but the range we are given is \((-鈭�,-2] \cup [2, 鈭�)\), which means we have a vertical stretch by a factor of 2. Therefore, the final function becomes \(y = 2csc((2/3)x)\).

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