Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Give a short answer to each question. If \(f(a)=-5,\) what is the value of \(|f(a)| ?\)

Step-by-Step Solution

Answer

The value of \(|f(a)|\) is 5.

Step 1: Understand Absolute Value

The absolute value of a number is the distance of the number from zero on the number line, without considering direction. It is always non-negative.

Step 2: Apply Absolute Value Definition

Given that \( f(a) = -5 \), the absolute value is determined by removing the negative sign. Hence, \( |f(a)| = |-5| \).

Step 3: Evaluate the Expression

Evaluate \( |-5| \): Since the absolute value of \(-5\) is 5, we have \( |f(a)| = 5 \).

Key Concepts

Functions

Functions are mathematical relationships where each input is related to exactly one output. Think of a vending machine: you press a button (input) and get a specific item (output). In mathematics, a function takes an input, operates according to a specific rule and produces an output. If we say \( f(a) = -5 \), \( f \) is the function and \( a \) is the input, resulting in the output \(-5\).

Functions are used to link variables and to describe patterns or relationships.

• **Function Notation:** Looks like \( f(x) \), where \( x \) is the input variable and \( f(x) \) is the output.
• **Domain:** The set of all possible inputs.
• **Range:** The set of all possible outputs.

Understanding functions helps you comprehend how one quantity depends on another, making them vital in everyday calculations and scientific experiments.

Number Line

The number line is a visual representation of numbers arranged in order. Imagine a straight line where numbers are placed at equal intervals, starting with negative numbers on the left, zero in the center, and positive numbers on the right.

This tool provides a sense of spacing and magnitude, allowing for easy visualization of addition, subtraction, and the concept of absolute value.

Here's how the number line helps in our exercise:

• **Zero Point:** Serves as the midpoint where the direction changes from negative to positive.
• **Negative Numbers:** Located to the left of zero; their absolute values are their positive counterparts.
• **Absolute Value:** Reflects distance from zero; for example, both \(-5\) and \(5\) have an absolute value of \(5\) because they are five units away from zero.

Using the number line, one can easily visualize why the absolute value of \(-5\) is \(5\), reinforcing your learning of the concept.

Distance

In mathematics, distance often refers to how far apart two points are on the number line. Absolute value is a way to measure this distance. Think of absolute value as the length of a journey, where the sign doesn't matter, only the magnitude of steps taken.

The beauty of absolute value is its application in measuring real-world distances and understanding differences, regardless of direction. Some key points include:

• **Non-Negative:** Distance is always positive or zero; just like absolute value.
• **Real-World Applications:** Used to measure things like temperature changes or financial gains/losses.
• **Formula Connection:** If \( f(a) = -5 \), \(|f(a)| = 5\) because distance is measured from zero on the number line.

Recognizing the connection between distance and absolute value can help you in various practical situations, enhancing both your math skills and problem-solving abilities.