Chapter 15: Problem 4

\(f(x)= \begin{cases}1, & \pi / 2<|x|<\pi \\ 0, & \text { otherwise }\end{cases}\)

Short Answer

Expert verified

The function outputs 1 for \(-\pi < x < -\pi/2\) or \(\pi/2 < x < \pi\), and 0 otherwise.

Step by step solution

- Understand the Function

The function given is a piecewise function. It is defined differently based on the value of \(|x|\). Specifically, it outputs 1 when \(\pi/2 < |x| < \pi\) and 0 otherwise.

- Analyze the Condition \(\pi/2 < |x| < \pi\)

To understand when the function outputs 1, solve the inequality \(\pi/2 < |x| < \pi\). This means that \(|x|\) (the absolute value of x) must be between \(\pi/2\) and \(\pi\).

- Translate the Absolute Value Condition

Since this inequality involves absolute value, consider two cases: \(-\pi < x < -\pi/2\) and \(\pi/2 < x < \pi\). These are the intervals where the function will return 1.

- Determine the Output for Other Values

For any value of x that does not fall in the intervals identified in Step 3, the function will output 0.

- Summarize the Piecewise Function

To summarize, \( f(x) = \begin{cases} 1, & -\pi < x < -\pi/2 \text{ or } \ \pi/2 < x < \pi \ 0, & \text{ otherwise} \ \end{cases} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value

First, let's discuss the concept of absolute value. The absolute value of a number, denoted as \(|x|\), is the distance of the number from zero on the number line. It is always a non-negative value. For example:

The absolute value of 3 is 3, written as \(|3| = 3\).
The absolute value of -3 is also 3, written as \(|-3| = 3\).

In our exercise, we deal with \(|x|\). The function is evaluated based on whether this value lies between \(\frac{\(\text{pi}\)}{2} \) and \(\text{pi}\). To find \(|x| < \text{pi}\), we consider x being in both the positive and negative ranges that fall within this bound, i.e., \(-\text{pi} < x < \text{pi}\).

Inequalities

Next, let's tackle inequalities. Inequalities express the relationship between two expressions indicating that one is greater than or less than the other. Common symbols include:

\(<\): less than
\(>\): greater than
\(\leq\): less than or equal to
\(\geq\): greater than or equal to

In our function, we deal with the inequality \(\frac{\text{pi}}{2} < |x| < \text{pi}\). To handle this, realize it needs two interpretations:

\(|x| > \frac{\text{pi}}{2}\): This includes values of \(|x|\) greater than \(\frac{\text{pi}}{2}\).
\(|x| < \text{pi}|\): And this includes values of \(|x|\) less than \(\text{pi}\).

Combining these, our concern is the interval between \(\frac{\text{pi}}{2} \) and \(\text{pi}\).

Function Analysis

Function analysis involves understanding how a function behaves over different intervals. For piecewise functions, each piece corresponds to a different set of conditions. Breaking down our example:

When \(\frac{\text{pi}}{2} < |x| < \text{pi}\): The function \(f(x)\) outputs 1.
For all other values of \(x\): The function \(f(x)\) outputs 0.

To thoroughly analyze:

Identify intervals meeting the conditions \(\frac{\text{pi}}{2} < |x| < \text{pi}\).
Determine the corresponding output for each interval.

When the interval condition \(\frac{\text{pi}}{2} < |x| < \text{pi}\) is met, the result is 1.
When not met, the result defaults to 0.