Absolute Value Function

The absolute value function is a mathematical function that gives the distance of a number from zero on the number line. It is denoted by |x|, where x is the input value.

The absolute value of a number is always non-negative. If the input value is positive or zero, then the absolute value is the same as the input value. If the input value is negative, then the absolute value is the opposite of the input value.

Properties of Absolute Value Function

1. Absolute Value of a Positive Number: |x| = x
2. Absolute Value of a Negative Number: |(-x)| = x
3. Absolute Value of Zero: |0| = 0
4. Absolute Value of a Sum: |x + y| ≤ |x| + |y|

Graph of Absolute Value Function

The graph of the absolute value function is a V-shaped graph, with the vertex at the origin (0, 0). It extends upwards and downwards from the vertex.

The equation of the absolute value function is y = |x|.

Examples

Let's consider a few examples to understand the absolute value function:

1. Find the absolute value of -5.

Answer: |(-5)| = 5

2. Solve the equation |2x - 3| = 7.

Answer: |2x - 3| = 7 can be re-written as 2x - 3 = 7 or 2x - 3 = -7. Solving these equations gives x = 5 or x = -2.

Study Guide

When studying the absolute value function, it's important to understand its properties, graph, and how to solve equations involving absolute values. Here are some key points to focus on:

1. Understand the concept of absolute value and its significance as the distance from zero.
2. Memorize the properties of the absolute value function.
3. Practice graphing the absolute value function and identifying its key characteristics such as the vertex and direction of the arms of the V-shaped graph.
4. Learn the methods for solving equations involving absolute values, such as isolating the absolute value and considering both the positive and negative cases.
5. Work on various problems and examples to strengthen your understanding of the absolute value function.

By mastering these concepts and practicing regularly, you'll be well-prepared to work with the absolute value function and its applications in mathematics.