Problem 80 - Precalculus with Calculus Previews

Question

Use a graphing utility to obtain the graph of the given function. Find all solutions of the equation \(f(x)=0\) and an interval that contains all these solutions. $$ f(x)=\cos \left(\frac{\pi}{2 x}\right) $$

Step-by-Step Solution

Answer

The solutions are of the form \(x = \frac{1}{2n+1}\) for odd \(n\), within the interval \((-1, 1)\).

Step 1: Understanding the Function

We are given the function \(f(x) = \cos\left(\frac{\pi}{2x}\right)\). To find where \(f(x) = 0\), we need to understand when the cosine function equals zero. Since cosine equals zero at odd multiples of \(\frac{\pi}{2}\), we need to find \(x\) such that \(\frac{\pi}{2x} = \left(n + \frac{1}{2}\right)\pi\) for odd integers \(n\).

Step 2: Setting up the Equation

To find when \(f(x) = 0\), set \(\frac{\pi}{2x} = \left(n + \frac{1}{2}\right)\pi\). Simplifying gives \(\frac{1}{2x} = n + \frac{1}{2}\), which results in \(x = \frac{1}{2n + 1}\).

Step 3: Finding Solutions

The solutions to \(f(x) = 0\) are now expressed in terms of odd integers as \(x = \frac{1}{2n+1}\). These solutions can be either positive or negative, corresponding to positive and negative values of \(n\).

Step 4: Determining the Interval Containing All Solutions

Observe that as \(|n|\) increases, \(x = \frac{1}{2n+1}\) approaches zero. The solutions for \(x\) will lie in the interval \((-1, 1)\) because \(-1 < \frac{1}{2n+1} < 1\) for all non-zero odd integers \(n\).

Step 5: Using a Graphing Utility

To graph the function \(f(x) \), use a graphing utility like Desmos, GeoGebra, or a similar tool. For the range of interest \(-1 < x < 1\), observe the points where the graph intersects the x-axis. These intersections confirm the solutions set as \(x = \frac{1}{2n+1}\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function, often abbreviated as "cos." It is an even function, meaning that it is symmetric around the y-axis. For any angle \(\theta\), the cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

The general form of the cosine function is \(y = \cos(x)\), which can be modified with transformations such as translations, stretches, and compressions. The given exercise utilizes a cosine function with a transformation:

• \(f(x) = \cos\left(\frac{\pi}{2x}\right)\)

This transformed function changes the period and domain, converging towards zero as \(x\) moves away from zero. Understanding these behavior changes is crucial for solving equations involving trigonometric functions.

Zeros of Trigonometric Functions

Zeros for trigonometric functions are values of \(x\) where the function equals zero. For the cosine function, these zeros occur at odd multiples of \(\pi/2\). In our exercise with the transformed function, we're interested in finding zeros for the equation \(f(x) = \cos\left(\frac{\pi}{2x}\right) = 0\).

To solve the equation, we set the transformed angle equal to any odd multiple of \(\pi/2\) :

• \(\frac{\pi}{2x} = (n + \frac{1}{2})\pi \)

Solving for \(x\) yields the zeros in terms of \(n\) by isolating \(x\):

• \(x = \frac{1}{2n + 1}\)

These solutions show the points at which the cosine function crosses the x-axis, revealing its periodic nature across different values of \(x\). This understanding is vital when analyzing and graphing trigonometric functions.

Interval Notation

Interval notation is a mathematical method used to specify a range of values. It expressively denotes from which points to which points a set extends. In our exercise, interval notation helps define where all solutions to \(f(x) = 0\) lie.

The given solutions \(x = \frac{1}{2n+1}\) converge toward zero as \(n\) increases. However, they do not actually include zero, as \(n\) must be non-zero for the function to be valid. Therefore, the valid interval for all these solutions is expressed as:

• \((-1, 1)\)

This interval excludes the endpoints, demonstrating that values within are approached but never reached by our function. Knowing how to express solutions in interval notation aids in both solving the problem and visually interpreting results using a graphing calculator.

Graphing Utilities

Graphing utilities are digital tools used to visualize mathematical functions and equations. They provide a way to see how trigonometric functions behave across specified domains.

Popular graphing utilities include:

• Desmos
• GeoGebra
• Graphing Calculators

Using these for our function \(f(x) = \cos\left(\frac{\pi}{2x}\right)\), allows us to observe and verify the function's zero points visually. By inputting the function and setting a domain, like \(-1 < x < 1\), one can see where the graph crosses the x-axis.

Graphing utilities not only confirm algebraic calculations but also show additional features like symmetry, periodicity, and proximity of zeros. These tools are invaluable for understanding complex trigonometric behavior and gaining deeper insights into mathematical functions.