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Chapter 4: Problem 96

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

Short Answer

Expert verified
The equation of the function is \(y = -\pi \cdot \csc{(\pi x)}\).

Step by step solution

01

Recall General Form for Cosecant Function

First, recall that the general form for the cosecant function is given by \(y = A \cdot \csc{(Bx + C)} + D\), where \(A\) is the amplitude, \(B\) determines the period, \(C\) causes horizontal shifts, and \(D\) causes vertical shifts.
02

Determine Coefficient for Period

We are told that the period of the function is 2. For cosine and sine functions, the period is determined by \(B\) in the general form, given as \(\frac{2\pi}{B}\), so \(B = \frac{2\pi}{\text{period}}\). Substituting our given period into this formula gives \(B = \frac{2\pi}{2} = \pi\). This means the equation of the function now becomes \(y = A \cdot \csc{(\pi x + C)} + D\).
03

Identify the Range and Adjust Function

From the problem, the range is given as \(-\infty,-\pi] \cup[\pi, \infty)\). The range of the cosecant function is usually \(-\infty,-1] \cup[1, \infty)\), but in this case, it can be achieved by multiplying the cosecant function by negative pi. This multiplies all the points on the function by negative pi, achieving our desired range. However, it also flips the function upside down. This results in the function \(y = -\pi \cdot \csc{(\pi x + C)} + D\).
04

Finalizing the Equation

Because a shift or a stretch doesn't change the range of the cosecant function, the constants \(C\) and \(D\) won't affect the range of the function. Therefore, the final equation is \(y = -\pi \cdot \csc{(\pi x)}\).

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