The Absolute Value Function: A Key Concept in Mathematics Explained

Absolute Value Function

The absolute value function is a fundamental concept in mathematics that represents the distance of a number from zero on the number line

The absolute value function is a fundamental concept in mathematics that represents the distance of a number from zero on the number line. It is denoted as |x|, where x can be any real number. The absolute value of x is always non-negative.

The absolute value function has a simple definition:
– If x is positive or zero, then |x| is equal to x.
– If x is negative, then |x| is equal to -x (the negative of x).

To understand the absolute value function graphically, we can consider a graph of |x|. The graph is a V-shaped figure, with the vertex at the origin (0, 0). It extends infinitely in both the positive and negative directions of the x-axis.

The key characteristic of the absolute value function is that it never produces a negative value. The output is always non-negative, regardless of the input. For example, |3| equals 3, |0| equals 0, and |-5| equals 5.

Another important property is that the absolute value function is always symmetric with respect to the y-axis. This means that if we reflect any point (x, y) on the graph of |x| about the y-axis, we will get the point (-x, y).

When working with absolute value functions, it is helpful to be familiar with a few rules:

1. |-a| = |a|: The absolute value of a negative number is equal to the absolute value of its positive counterpart. For example, |-3| equals 3 and |(-2)| equals 2.

2. |a * b| = |a| * |b|: The absolute value of the product of two numbers is equal to the product of their absolute values. For example, |2 * 3| equals 6 and |(-5) * 4| equals 20.

3. |a + b| ≤ |a| + |b|: The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values. This property is known as the Triangle Inequality. For example, |2 + 3| is less than or equal to |2| + |3|, which simplifies to 5 ≤ 5.

Understanding the absolute value function is crucial not only in algebra but also in other areas of mathematics, such as calculus and linear programming. It is a versatile tool that helps solve various mathematical problems and analyze real-life situations.

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