Question

Graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=-2 \csc (\pi x) $$

Short answer

Domain: \( x \in \mathbb{R} \setminus \{k : k \text{ is an integer}\}\). Range: \((-\infty, -2] \cup [2, \infty)).\)

Step-by-step solution

Understand the Function

The given function is \(y = -2 \csc(\pi x)\). Recall that the cosecant function \(csc(x)\) is the reciprocal of the sine function \( \sin(x)\), so \(\csc(x) = \frac{1}{\sin(x)}\). Hence, this function has vertical asymptotes where \(\sin(\pi x) = 0\), i.e., at \(x = k\) for any integer \(k\).

Identify Key Points and Period

The function \(\csc(\pi x)\) has a periodicity of \(2\). Since \(\csc(\pi x)\) is undefined where \(\sin(\pi x) = 0\), its key points are found in the intervals between these vertical asymptotes. Therefore, \(\csc(\pi x)\) has a period of \(2\).

Find and Plot Key Points

To graph one period from \(-1\) to \(1\), find key points at x-values \(\frac{-1}{2}, \frac{1}{2} \) corresponding to the peaks and valleys of the sine function. Calculate the values: \(y=-2\csc(-\frac{\pi}{2}) = 2\) at \( x = -0.5\); \(y = -2\csc(\frac{\pi}{2}) = -2\) at \(x = 0.5\).

Graph the Function

Using the points and asymptotes, sketch the graph of \(y=-2\csc(\pi x)\). Label the vertical asymptotes at \(x = k\). Identify periods: plot points for a second cycle from \(1\) to \(3\), and repeat key points and asymptotes.

Determine Domain and Range

The domain of the function excludes any \(x = k\) for integers \(k\) where the function is undefined. Therefore, the domain is \(x \in \mathbb{R} \setminus \{k : k \text{ is an integer}\}\). The range includes all real numbers except between \(-2\) and \(2\), hence the range is \(( - \infty, -2 ] \cup [ 2, \infty )\).

Key concepts

Cosecant Function

The cosecant function, denoted as \(\text{csc}(x)\), is the reciprocal of the sine function. This means that \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\). Because it relies on the sine function, it has unique characteristics:

• The cosecant function is undefined where the sine function is zero, causing vertical asymptotes.
• Its values are large when the sine function values are small, and vice versa.
• To graph the function, you need to be careful of the points where the sine function crosses zero.

Understanding these properties helps in sketching and analyzing the graph of \(\text{csc}(x)\).

Periodicity

Periodicity refers to the repeating nature of trigonometric functions. For the cosecant function \(\text{csc}(\text{π} x)\), the period is closely tied to the sine function’s period. Since \(\text{sin}(πx)\) has a period of 2 (it completes a full cycle in 2 units), \(\text{csc}(πx)\) also repeats every 2 units.

• Within each period, the function will show all its characteristics: it will have maximum values, minimum values, and asymptotes.
• When graphing, indicating at least two periods helps in understanding the function’s recurring behavior.

By recognizing the periodicity, we can predict the behavior of the graph and easily sketch multiple cycles.

Vertical Asymptotes

Vertical asymptotes are lines that the function approaches but never touches or crosses. For the cosecant function \(\text{y} = -2 \text{csc}(πx)\), they occur where the sine function is zero because \(\text{csc}(x)\) is undefined at these points. In \(\text{y} = -2 \text{csc}(π x)\), this happens at integer values of x (where \(\text{π} x = kπ\) for any integer k).

• Thus, vertical asymptotes are at \(x = k\) for any integer k.
• They significantly impact the graph’s shape as the function climbs towards positive or negative infinity near these lines.

Plotting these asymptotes correctly helps you visualize and sketch the function accurately.

Domain and Range

The domain and range of a function describe where the function is defined and what values it can take. For \(\text{y} = -2 \text{csc}(π x)\), understanding these aspects is crucial.
The domain excludes points where the function is undefined, which are at integer values of x (due to the vertical asymptotes). Thus, the domain is \(x ∈ \text{ℝ} \backslash \text{k}\), where k is any integer.
The range specifies the output values. For \(\text{y} = -2 \text{csc}(π x)\), the function value will reach values far from zero and closer to infinity as it moves away from the vertical asymptotes. Hence, the range is \((-∞, -2] ∪ [2, ∞)\), avoiding the gap between -2 and 2. This tells us that the function never touches or falls between -2 and 2.