Problem 35 - Algebra for College Students

Question

\(f(x)=x-|x|\)

Step-by-Step Solution

Answer

\(f(x) = \begin{cases} 0 & \text{if } x \geq 0 \\ 2x & \text{if } x < 0 \end{cases}\)

Step 1: Understand the Absolute Value Function

The absolute value function, denoted by \(|x|\), returns the non-negative value of \(x\). If \(x\) is negative, \(|x| = -x\), and if \(x\) is positive or zero, \(|x| = x\).

Step 2: Consider the Piecewise Definition of Absolute Value

Given the function \(f(x) = x - |x|\), we can break it into cases based on the definition of \(|x|\):1. If \(x \geq 0\), then \(|x| = x\), leading to: \[f(x) = x - x = 0\]2. If \(x < 0\), then \(|x| = -x\), leading to:\[f(x) = x - (-x) = x + x = 2x\]

Step 3: Write the Piecewise Function

From Step 2, we can now write \(f(x)\) as a piecewise function:\[f(x) = \begin{cases} 0 & \text{if } x \geq 0 \ 2x & \text{if } x < 0\end{cases}\]

Step 4: Validate the Result

Now, verify that the function behaves as expected for sample values:- For \(x = 2\), \(x \geq 0\), so \(f(x) = 0\).- For \(x = -2\), \(x < 0\), so \(f(x) = 2(-2) = -4\).These sample calculations match our piecewise function definition.

Key Concepts

Understanding the Absolute Value

The absolute value function plays a crucial role in mathematics, symbolized by \(|x|\). This function extracts the non-negative essence from a number. Imagine it like a ruler that measures only the distance, ignoring the direction.
Here's how it functions:

• If the number is positive, or zero, it stays the same.
• If the number is negative, it turns positive by flipping its sign.

For example, \(|3| = 3\) and \(|-3| = 3\).
So, regardless of the sign of the number within, the absolute value aims to give you a 'positive-only' reality of what it's dealing with. This characteristic is foundational for breaking down functions like \(f(x) = x - |x|\), where signs dictate the output.

Unraveling Function Behavior

When analyzing a function like \(f(x) = x - |x|\), it's essential to understand how it behaves across different domains of its input. The behavior is determined by the absolute value, and hence the piecewise definition. Behavior Breakdown:

• Non-Negative Inputs (\(x \geq 0\)): Here, the absolute value part matches the input, \(|x| = x\). Thus, \(f(x) = x - x = 0\). Notice that for any non-negative \(x\), the result is consistently zero.
• Negative Inputs (\(x < 0\)): Here, the absolute value part counters the input with an opposite sign, \(|x| = -x\). Thus, \(f(x) = x - (-x) = 2x\). For these \(x\), the function doubles the value of \(x\).

This dual behavior based on input showcases how piecewise functions, with the help of absolute values, adapt function rules for different intervals.

Crafting a Piecewise Definition

To articulate the behavior of \(f(x) = x - |x|\), we establish a piecewise definition. Piecewise functions are especially useful for defining rules that depend on conditions. Here, those conditions are the sign of \(x\).
A Piecewise Structure:1. Define conditions based on the domain of input: - If \(x \geq 0\), apply the rule where \(f(x) = 0\). - If \(x < 0\), apply the rule where \(f(x) = 2x\).2. Express the above setup in a clear mathematical form: \[f(x) = \begin{cases} 0 & \text{if } x \geq 0 \ 2x & \text{if } x < 0 \end{cases}\]Such formulations make the behavior of the function easily understandable at a glance. They allow quick comprehension of how different domains affect output, ensuring that every potential input has a precise rule to follow.