Understanding the Absolute Value Function: Definition, Examples, and Properties

absolute value function

The absolute value function, denoted as |x|, is a mathematical function that gives the distance of a number from zero on the number line

The absolute value function, denoted as |x|, is a mathematical function that gives the distance of a number from zero on the number line. It returns the non-negative value of a given input regardless of its sign.

The absolute value function defines the output (y) as the distance between the input (x) and zero. If x is positive, then |x| = x. If x is negative, then |x| = -x.

Here are a few examples:

Example 1: |3| = 3
In this case, since 3 is positive, the absolute value of 3 is equal to 3 itself.

Example 2: |-5| = 5
Here, since -5 is negative, we take the opposite sign of -5, which is 5. So, the absolute value of -5 is 5.

Example 3: |0| = 0
Zero is the only number that has an absolute value of zero since it is equidistant from both the positive and negative numbers on the number line.

The absolute value function has several properties:

1. Non-negativity: The absolute value of a number is always non-negative or zero.

2. Symmetry: The absolute value function is symmetric about the y-axis since the absolute value of any positive number is the same as the absolute value of its negative counterpart.

3. Triangle Inequality: The absolute value function satisfies the triangle inequality property, which states that for any two numbers a and b, |a + b| ≤ |a| + |b|. This means that the absolute value of the sum of two numbers is always less than or equal to the sum of their absolute values.

The absolute value function is often used in various areas of mathematics, including solving equations, calculating distances, and representing magnitudes. It is an essential concept across different branches of mathematics and has wider applications in real-life scenarios as well.

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