Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\frac{1}{2} \csc x$$

Step-by-Step Solution

Answer

(a) Period: \( 2\pi \); (b) Phase shift: 0; (c) Range: \(( -\infty, -\frac{1}{2} ] \cup [ \frac{1}{2}, \infty )\)."

Step 1: Identify the Function Type

The given function is \( y = \frac{1}{2} \csc x \). The \( \csc \) function is the cosecant function, which is the reciprocal of the sine function, \( \csc x = \frac{1}{\sin x} \).

Step 2: Determine the Period

The basic period of \( \csc x \) is the same as that of \( \sin x \), which is \( 2\pi \). There are no modifications to the \( x \) within the cosecant function that would affect the period. Hence, the period remains \( 2\pi \).

Step 3: Examine the Phase Shift

The phase shift is determined by any horizontal shifts represented inside the function. Since the equation \( \frac{1}{2} \csc x \) does not contain any terms of the form \( x - c \) inside the cosecant function, there is no phase shift. Thus, the phase shift is 0.

Step 4: Calculate the Range

The range of \( \csc x \) is the set of all real numbers except those between -1 and 1 (i.e., \(( -\infty, -1 ] \cup [ 1, \infty )\)). Multiplying by \( \frac{1}{2} \) scales these values by the same factor, so the range of \( \frac{1}{2} \csc x \) becomes \(( -\infty, -\frac{1}{2} ] \cup [ \frac{1}{2}, \infty )\).

Key Concepts

Period of a Function

In trigonometry, the period of a function refers to the distance over which the function's graph repeats itself. Understanding the period helps us predict the behavior of the function over its domain.
Let's dive into the specifics for the cosecant function. Since cosine and sine functions are the most familiar periodic functions, it is helpful to start there. The cosecant function, noted as \( \csc x \), is directly related to the sine function as it is its reciprocal. Therefore, it inherits the period of the sine function.

• The basic period of both the sine, \( \sin x \), and cosecant, \( \csc x \), is \( 2\pi \).
• The function \( y = \frac{1}{2} \csc x \) does not include any alterations to \( x \) (like multiplication or addition within the argument), thus keeping the period unchanged.

To sum up, the graph of \( \frac{1}{2} \csc x \) will repeat every \( 2\pi \) units along the x-axis.

Phase Shift

The phase shift of a trigonometric function is the horizontal translation of its graph. We often denote this as the "shift" of the wave along the x-axis.
When you're examining how the phase shift works, look for changes inside the parenthesis of the function's argument, typically seen in expressions like \( \sin(x - c) \) or \( \cos(x + c) \). This "c" value represents how far the graph shifts; however, its sign changes the direction (i.e., \( x - c \) shifts to the right, \( x + c \) shifts to the left).

• In the function \( y = \frac{1}{2} \csc x \), there are no such shifts or alterations. The variable \( x \) appears without any additional terms inside the function.
• Therefore, the phase shift is considered to be 0 for this particular cosecant function.

Overall, there is no phase shift, meaning the graph starts its cycle without any horizontal displacement.

Range of a Function

The range of a function refers to the set of all possible output values (y-values) the function can produce. For many trigonometric functions, the range is a bit more restricted due to their periodic nature.
When we look at \( y = \csc x \), we understand immediately that this function is reciprocal to \( \sin x \). This relationship affects the range significantly because the sine function only outputs values between -1 and 1.

• The basic range of the cosecant function is all real numbers except those between -1 and 1. In notation, this is \( (-\infty, -1] \cup [1, \infty) \).
• By multiplying the \( \csc x \) by \( \frac{1}{2} \), the range of \( y = \frac{1}{2} \csc x \) scales accordingly.

The scaled range becomes \( (-\infty, -\frac{1}{2}] \cup [\frac{1}{2}, \infty) \). So, the values of \( \frac{1}{2} \csc x \) will fall either below \(-\frac{1}{2}\) or above \(\frac{1}{2}\), never in between these numbers.