Problem 22 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. Indicate any intercepts and symmetry, and determine whether the function is even, odd, or neither. $$ 2-\csc x $$

Step-by-Step Solution

Answer

The function is neither even nor odd, with x-intercepts at \( \frac{\pi}{6} + 2n\pi \) and \( \frac{5\pi}{6} + 2n\pi \).

Step 1: Understand the Function

The given function is \( f(x) = 2 - \csc(x) \). The cosecant function \( \csc(x) \) is the reciprocal of the sine function, meaning \( \csc(x) = \frac{1}{\sin(x)} \). The function \( f(x) = 2 - \csc(x) \) will have vertical asymptotes wherever \( \csc(x) \) is undefined, i.e., where \( \sin(x) = 0 \). These points occur at \( x = n\pi \) for any integer \( n \).

Step 2: Find the Intercepts

To find the y-intercept, evaluate \( f(0) = 2 - \csc(0) \). Since \( \csc(0) = \frac{1}{0} \) (undefined), there is no y-intercept. For x-intercepts, set the function equal to zero: \( 2 - \csc(x) = 0 \). Solving for \( x \) gives \( \csc(x) = 2 \), leading to \( \sin(x) = \frac{1}{2} \). The solutions to this are \( x = \frac{\pi}{6} + 2n\pi \) and \( x = \frac{5\pi}{6} + 2n\pi \) for any integer \( n \).

Step 3: Determine the Symmetry

Check if the function is even, odd, or neither. A function is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \). Calculate \( f(-x) = 2 - \csc(-x) = 2 + \csc(x) \). Since \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \), the function is neither even nor odd.

Step 4: Sketch the Graph

Use the information gathered: plot vertical asymptotes at \( x = n\pi \). Mark x-intercepts at \( x = \frac{\pi}{6} + 2n\pi \) and \( \frac{5\pi}{6} + 2n\pi \). Sketch the basic shape of \( \csc(x) \), which involves moving the graph of \( -\sin(x) \) upward by 2 units for \( 2 - \csc(x) \). These movements should reflect that \( f(x) \) approaches 2 as \( \sin(x) \) approaches infinity or negative infinity between the asymptotes.

Step 5: Analyze and Conclude

We've determined the intercepts, asymptotes, and symmetry of the function, and we've sketched the graph. Intercepts appear at \( x = \frac{\pi}{6} + 2n\pi \) and \( x = \frac{5\pi}{6} + 2n\pi \), and there is no y-intercept. The function is neither even nor odd.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial for grasping the behavior of trigonometric functions. In trigonometry, one of the basic identities involves the sine and cosecant functions:

• \( ext{sin}(x) \) is the opposite of \( ext{cosecant}(x) \); specifically \( ext{csc}(x) = \frac{1}{\text{sin}(x)} \).

This relationship tells us that wherever \( ext{sin}(x) \) is zero, \( ext{csc}(x) \) becomes undefined, leading to vertical asymptotes in the graph. For example, \( ext{sin}(x) = 0 \) at multiples of \( \pi \), specifically at \( x = n\pi \) for integer \( n \). These points are key to determining the behavior and sketching graphs of functions involving \( \text{csc}(x) \).

Additionally, trigonometric identities can assist in finding x-intercepts. In our function, \( 2 - \text{csc}(x) \) is set to zero to find when the graph crosses the x-axis. By equating \( \text{csc}(x) = 2 \), we derive that \( \text{sin}(x) = \frac{1}{2} \), guiding us to specific x-values where this condition holds. This connection between \( \text{csc}(x) \) and \( \text{sin}(x) \) is an excellent example of how identities simplify graphing complex trigonometric functions.

Function Symmetry

Exploring function symmetry helps clarify whether a given trigonometric function is even, odd, or neither.

An even function satisfies \( f(-x) = f(x) \) which means its graph is symmetrical about the y-axis. Conversely, an odd function meets the condition \( f(-x) = -f(x) \), indicating symmetry about the origin. Understanding these properties simplifies graph analysis and helps in identifying specific symmetry in a graph.

• If \( f(-x) = f(x) \), the function is even.
• If \( f(-x) = -f(x) \), the function is odd.
• If neither condition is fulfilled, the function is neither even nor odd.

For the function \( f(x) = 2 - \text{csc}(x) \), substituting \( -x \) gives \( f(-x) = 2 + \text{csc}(x) \), not equal to \( f(x) \) nor \(-f(x) \). Therefore, it is classified as neither even nor odd.

Understanding symmetry is vital when predicting the graph's general shape. For example, knowing that a function is not symmetric simplifies graph composition by allowing for less repetitive plotting. You can focus on plotting sections between known critical points like intercepts and asymptotes.

X-Intercepts of Trigonometric Functions

Finding x-intercepts is an integral step in graphing trigonometric functions. An x-intercept occurs where a function crosses the x-axis, meaning the function's value is zero at these points.

To locate x-intercepts for \( f(x) = 2 - \text{csc}(x) \), you set the equation \( 2 - \text{csc}(x) = 0 \). Solving this, you determine \( \text{csc}(x) = 2 \), which implies \( \text{sin}(x) = \frac{1}{2} \). Understanding when \( \text{sin}(x) \) equals \( \frac{1}{2} \) is crucial. Using the unit circle:

• \( x = \frac{\pi}{6} + 2n\pi \)
• \( x = \frac{5\pi}{6} + 2n\pi \)

for any integer \( n \), are where the sine function produces the value \( \frac{1}{2} \).

Recognizing these specific angles ensures accuracy in graph placement and makes it easier to anticipate where intercepts occur. Additionally, these calculations aid in understanding the periodic nature of trigonometric functions. Knowing the intervals or pattern of x-intercepts can offer insight into the function's symmetry and periodicity as well.