The Absolute Value Function: Properties, Definitions & Applications

absolute value function

a function that contains an absolute value expression

The absolute value function is a mathematical function that takes a real number as input and produces the magnitude of that number as output. In other words, it returns the distance of a number from zero on the number line. The absolute value function is denoted using vertical bars around the input: |x|. For example, |5| is equal to 5, and |-3| is equal to 3.

The absolute value function has some important properties, including:

1. Non-negativity: The output of the function is always non-negative or equal to zero. This means that |x| >= 0 for all values of x.

2. Symmetry: The function is symmetric around the y-axis. This means that |x| = |-x| for all values of x.

3. Piecewise definition: The function can be defined in a piecewise manner, as follows:

* If x is positive, then |x| = x.
* If x is negative, then |x| = -x.
* If x is zero, then |x| = 0.

The absolute value function is used in a variety of mathematical contexts, including solving equations involving absolute values, finding the distance between two points in a coordinate plane, and defining the norm of a vector in linear algebra.

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