Problem 88 - Intermediate Algebra

Question

What is the name of the function \(f(x)=|x| ?\)

Step-by-Step Solution

Answer

The name of the function is the "absolute value function."

Step 1: Understanding Absolute Value

The absolute value function, denoted as \(|x|\), is defined as the distance of a number from zero on the number line, without considering the direction. It is always non-negative.

Step 2: Definition of the Given Function

The expression \(f(x) = |x|\) defines a function where for any real number \(x\), \(f(x)\) is the absolute value of \(x\). This means \(f(x) = x\) if \(x\geq0\) and \(f(x) = -x\) if \(x<0\).

Step 3: Identifying the Function Name

Recognize that the name of the function given by \(f(x) = |x|\), which outputs the absolute value of the input, is commonly referred to as the "absolute value function."

Key Concepts

Absolute Value

The absolute value of a number is a measure of how far that number is from zero on the number line. It is denoted by two vertical bars around the number, such as \(|x|\). The absolute value is always a non-negative number, regardless of whether the original number is positive or negative.

• If a number is positive or zero, its absolute value is the same number.
• If a number is negative, its absolute value is the positive counterpart of that number.

For example:

• \(|3| = 3\)
• \(|-3| = 3\)

By focusing on the distance rather than direction, absolute values provide a consistent measure of magnitude.

Function Definition

Functions are mathematical constructs that assign a unique output for each valid input. The function \(f(x) = |x|\) is a well-known example, often referred to as the "absolute value function." ### How Does This Function Work?The absolute value function takes any real number \(x\) and returns its absolute value. Here's how it processes the input:

• If \(x\) is greater than or equal to zero, then \(f(x) = x\).
• If \(x\) is less than zero, then \(f(x) = -x\), which makes the result positive.

This piecewise definition ensures \(f(x)\) is always non-negative, aligning with the absolute value concept.

Number Line

The number line is a visual representation that helps us understand the position and value of numbers, as well as their absolute values. It is a straight line where each point corresponds to a real number, with zero as the central point.### Using the Number Line for Absolute ValueThe number line makes it easier to comprehend absolute value. Each number's absolute value is its distance from zero:

• Positive numbers are naturally to the right of zero.
• Negative numbers are to the left of zero, but their absolute value considers distance in a positive magnitude.

For instance, the number \(-4\) lies four units left of zero. Its absolute value is \(4\), because it is four units away, emphasizing the concept of distance rather than direction on the number line. This intuitive representation aids in grasping how absolute values function in real-world applications.