Problem 17 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(f(x) = 0\) for \(x \geq 0\); \(f(x) = 2x\) for \(x < 0\).

Step 1: Understanding the Absolute Value

The function contains an absolute value, \(|x|\), which is defined as \(|x| = x\) when \(x \geq 0\) and \(|x| = -x\) when \(x < 0\). This means the absolute value function will split our function into two cases based on the sign of \(x\).

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

For \(x \geq 0\), the absolute value \(|x| = x\). Thus, the function simplifies to:\[ f(x) = x - |x| = x - x = 0 \]So, for all non-negative \(x\), \(f(x) = 0\).

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

For \(x < 0\), the absolute value \(|x| = -x\). Thus, the function becomes:\[ f(x) = x - |x| = x - (-x) = x + x = 2x \]So, for all negative \(x\), \(f(x) = 2x\).

Step 4: Combining Both Cases

Now we combine the results of both cases: \[ f(x) = \begin{cases} 0, & \text{if } x \geq 0 \ 2x, & \text{if } x < 0 \end{cases} \]This piecewise function summarizes the behavior of \(f(x)\) based on the sign of \(x\).

Key Concepts

Piecewise Function

A piecewise function is a type of function characterized by different expressions for different intervals of the input variable. **These intervals usually depend on some conditions, such as the value of the input variable.**

In mathematics, unlike regular functions, piecewise functions can have different rules or expressions based on the input values.

• They are written using different cases, each with its corresponding condition.
• Commonly, there are at least two cases, often split by a specific value of the input.

In this exercise, the function \( f(x) = x - |x| \) is expressed as a piecewise function:

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

This notation shows clearly how the expression for \( f(x) \) changes depending on whether \( x \) is non-negative or negative. Piecewise functions are very useful in representing situations where rules change based on certain conditions, like tax brackets or speed limits.

Function Behavior

Function behavior refers to the changes in the function's output as its input varies. It helps to understand how the function reacts to different input values, and what patterns or properties can be observed.

• One key aspect of function behavior is identifying intervals where the function has specific expressions or rules, like in a piecewise function.
• This can involve determining when the function is increasing, decreasing, constant, or when it exhibits symmetry.

In our absolute value function \( f(x) = x - |x| \), the behavior differs drastically based on whether \( x \) is positive or negative:

• When \( x \geq 0 \), the function remains constant at 0. This means it does not increase or decrease as long as \( x \) is non-negative.
• When \( x < 0 \), the function increases as \( x \) becomes less negative, since \( f(x) = 2x \) leads the value to be negative but smaller as \( x \) approaches 0 from the left.

Understanding these aspects can greatly aid in visualizing and sketching the graph of the function, highlighting key features and any critical points.

Algebraic Expressions

Algebraic expressions form the backbone of mathematical problem-solving and are crucial in constructing and simplifying functions. **They consist of variables, numbers, and arithmetic operations.** In our function \( f(x) = x - |x| \), the expression involves a combination of a linear term and an absolute value.

Understanding and manipulating algebraic expressions enable us to break down complex problems into simpler parts.

• The expression \( x - |x| \) captures how subtraction of an absolute value affects the behavior of the function, leading to a transformation into a piecewise format.
• Algebraic techniques involve simplifying expressions by combining like parts and managing operations such as addition, subtraction, and factoring.

By separating the function into its piecewise parts using algebraic methods, we gain clarity on how each segment of the expression contributes to the overall function. This is especially helpful in identifying intervals and simplifying expressions within each interval, allowing for easier interpretations and calculations.