Problem 34 - Algebra for College Students

Question

\(f(x)=x+|x|\)

Step-by-Step Solution

Answer

The function is piecewise: \(f(x) = 2x\) for \(x \geq 0\) and \(f(x) = 0\) for \(x < 0\).

Step 1: Identify the Absolute Value Condition

To solve the function \( f(x) = x + |x| \), begin by understanding the behavior of the absolute value. The absolute value function \( |x| \) has different expressions based on whether \( x \) is positive or negative. Specifically, \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). We will consider two cases based on this condition.

Step 2: Case 1 - When \(x \geq 0\)

For \(x \geq 0\), the absolute value function \(|x|\) simplifies to \(x\). Thus, the function becomes: \[f(x) = x + x = 2x\]For all values of \(x\) that are greater than or equal to zero, \(f(x) = 2x\).

Step 3: Case 2 - When \(x < 0\)

For \(x < 0\), the absolute value function \(|x|\) becomes \(-x\). Hence, the given function modifies to: \[f(x) = x + (-x) = 0\]For all \(x < 0\), the function results in \(f(x) = 0\).

Step 4: Combine the Cases to Define the Piecewise Function

Combining both cases, the function \(f(x) = x + |x|\) can be defined as a piecewise function:\[f(x) = \begin{cases} 2x, & \text{if } x \geq 0 \0, & \text{if } x < 0 \end{cases}\]

Key Concepts

Absolute Value

Understanding the absolute value is crucial when working with piecewise functions. The absolute value of a number \(x\), denoted as \(|x|\), represents the distance from zero on the number line, regardless of direction. This means:

• For positive or zero values of \(x\), \(|x| = x\).
• For negative values of \(x\), \(|x| = -x\).

This dual nature is what makes the function \(f(x) = x + |x|\) interesting since it changes behavior depending on whether \(x\) is positive or negative. By recognizing these conditions, we can break problems into manageable parts. This approach aids in dissecting complex functions into simpler expressions.

Function Behavior

The behavior of any function changes based on its inputs. Specifically, a piecewise function like \(f(x) = x + |x|\) adjusts based on the value of \(x\). In this case:

• When \(x \geq 0\), the absolute value has no effect as it equals \(x\). The function simplifies to \(f(x) = 2x\).
• When \(x < 0\), the absolute value converts \( -x \) so adds up to zero. Therefore, the function remains \(f(x) = 0\).

This function shifts dynamically from one expression to another based on the sign of \(x\). Observing how the function adapts to positive and negative inputs demonstrates essential behaviors in piecewise functions.

Case Analysis

Case analysis involves examining distinct scenarios separately to understand the complete function behavior. With \(f(x) = x + |x|\), we consider two cases based on the value of \(x\):

• Case 1: if \(x \geq 0\), solve \(f(x)\) as \(2x\).
• Case 2: if \(x < 0\), solve \(f(x)\) as \(0\).

By evaluating each condition, we develop a comprehensive understanding. This step-by-step, case-by-case methodology helps define piecewise functions effectively. It also enables us to articulate the function's complete representation, as seen in the final piecewise function:\[ f(x) = \begin{cases} 2x, & \text{if } x \geq 0 \0, & \text{if } x < 0 \end{cases} \] Applying case analysis ensures precision in solving problems involving piecewise functions, as one considers every relevant condition and its influence on the function.