The Absolute Value Function: Definition, Characteristics, And Applications

Absolute Value Function

f(x) = |x|

The absolute value function is a mathematical function that returns the distance of a number from zero without considering its sign. It is represented by two vertical bars (|x|) and is defined as:

| x | = x if x ≥ 0
| x | = -x if x < 0 For example, the absolute value of -4 is 4, because its distance from zero on the number line is 4. Likewise, the absolute value of 5 is 5 because its distance from zero is also 5. Some characteristics of the absolute value function include: 1. Symmetry: The absolute value function is symmetric with respect to the y-axis. This means that if you reflect the graph of the function across the y-axis, you will get the same graph. 2. Piecewise Defined: The absolute value function is piecewise defined since its definition consists of two cases, one for values greater or equal to zero and one for values less than zero. 3. Always Positive: The absolute value of any number is always a positive number or zero. 4. Differentiability: The absolute value function is not differentiable at x = 0 because it has a sharp corner at that point. The absolute value function has many applications in various fields such as physics, economics, statistics, and computer science. It is commonly used to determine the magnitude of a vector, as well as to express the distance between two points in coordinate geometry.

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