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

Absolute Value Function

The absolute value function, denoted by |x|, is a mathematical function that returns the absolute value (or magnitude) of a real number

The absolute value function, denoted by |x|, is a mathematical function that returns the absolute value (or magnitude) of a real number. The absolute value of a number is its distance from zero on the number line, regardless of its sign.

The absolute value function can be defined as follows:
For any real number x,
– If x is positive or zero, then |x| = x.
– If x is negative, then |x| = -x.

This definition ensures that the absolute value function always returns a non-negative value. For example, |5| = 5 and |-5| = 5, since both 5 and -5 have a distance of 5 units from zero.

The absolute value function can be visualized as V-shaped graph that opens upwards. The vertex of the V-shaped graph is at (0, 0), representing the point where the function equals zero.

Here are a few properties of the absolute value function:
1. Non-negativity: Since the absolute value of any number is non-negative, |x| ≥ 0 for all real numbers x.
2. Symmetry: The absolute value function is symmetric about the y-axis. This means that |x| = |-x| for all real numbers x.
3. Triangle inequality: For any two real numbers a and b, |a + b| ≤ |a| + |b|. This inequality states that the absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.

The absolute value function has various applications in mathematics and science. It is commonly used to compute distances, find solutions to equations involving absolute values, and model real-world scenarios. For example, in physics, the absolute value function is used to calculate the magnitude of vectors.

To summarize, the absolute value function is a mathematical function that returns the non-negative value (or magnitude) of a real number. It has specific properties, including non-negativity, symmetry, and the triangle inequality. Understanding the absolute value function is fundamental in various mathematical concepts and applications.

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