As a domain expert in mathematics, I am well-versed in various mathematical concepts and functions. One of the fundamental concepts in algebra is the absolute value function. Let's delve into its definition and properties.
Absolute Value Function Definition:
The absolute value function is a mathematical function that measures the magnitude or distance of a number from zero on the number line, regardless of the direction. It is denoted by the symbol `| |` and is used to define a function that contains an algebraic expression within these symbols. The absolute value of a number is always non-negative.

Notation:
The absolute value function is generally written as `f(x) = |x|`, where `x` is the variable representing the input to the function.

Properties:

1. Non-Negativity: The absolute value of any real number is always greater than or equal to zero. This is expressed as `f(x) ≥ 0` for all real numbers `x`.

2. Identity: The absolute value of a positive number is the number itself, and the absolute value of a negative number is its opposite. This means `|x| = x` if `x ≥ 0` and `|x| = -x` if `x < 0`.

3. Symmetry: The absolute value function is an even function, which means `f(x) = f(-x)` for all `x`. This reflects the symmetry about the y-axis.

4. Linearity: The absolute value function is not linear because it does not satisfy the property of linearity, which states that for any real numbers `a` and `b`, `f(ax + b) = af(x) + b`.

5. Continuity: The absolute value function is continuous everywhere on the real number line. It has no breaks or gaps.

6. Differentiability: The absolute value function is differentiable everywhere except at `x = 0`. At `x = 0`, it has a corner point, which is a non-differentiable point.

Graph:
The graph of the absolute value function `f(x) = |x|` is a V-shaped curve. It starts at the origin (0,0), goes up to (1,1) when `x` is positive, and down to (-1,1) when `x` is negative. It reflects the behavior of the function at `x = 0`.

Applications:
Absolute value functions are used in various real-world contexts where the magnitude of a quantity is important, regardless of its direction. For example, they are used in calculating the distance between two points on the number line, in physics to measure the magnitude of vectors, and in business to represent the amount of a loss or gain, where the absolute value ensures that the result is non-negative.

Equations and Inequalities:
When solving equations or inequalities involving absolute values, it is often necessary to consider the cases where the value inside the absolute value is positive or negative separately. This is because the absolute value itself does not provide information about the direction of the original number.

Conclusion:
The absolute value function is a fundamental concept in mathematics that provides a way to measure the magnitude of a number without considering its sign. It is characterized by its non-negativity, symmetry, continuity, and differentiability (except at the origin). Understanding the properties and graph of the absolute value function is crucial for solving mathematical problems and for its applications in various fields.

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Absolute Value Functions. An absolute value function is a function that contains an algebraic expression within absolute value symbols. Recall that the absolute value of a number is its distance from on the number line. The absolute value parent function, written as f ( x ) = | x | , is defined as.read more >>