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

Absolute Value Function

The absolute value function is a mathematical function that gives the absolute value or the magnitude of a number

The absolute value function is a mathematical function that gives the absolute value or the magnitude of a number. It is denoted by the symbol |x|. The absolute value of a number is the distance of that number from zero on a number line.

The mathematical definition of the absolute value function can be given as follows:
For any real number x, the absolute value of x is defined as:
|x| =
x, if x is greater than or equal to zero,
or
-x, if x is less than zero.

For example, let’s take some values and calculate their absolute values:
1. |5| = 5
Here, the input number is 5, which is greater than zero, so we take the number itself.

2. |-3| = 3
Here, the input number is -3, which is less than zero, so we change the sign to positive and take its absolute value.

Properties of the absolute value function:
1. Non-Negativity: The absolute value of any number is always non-negative. This means that |x| ≥ 0 for any real number x.
2. Symmetry: The absolute value function has symmetry about the y-axis. This means that |x| = |-x|.
3. Triangle Inequality: The absolute value function satisfies the triangle inequality. This states that for any two real numbers x and y, |x + y| ≤ |x| + |y|.

Graphical representation:
The graph of the absolute value function is a V-shape, known as the absolute value graph or the “V” graph. It is symmetric about the y-axis. When graphing the absolute value function, it is helpful to pick some points to plot and connect them with straight lines.

Applications of the absolute value function:
1. Distance: The absolute value function is commonly used to calculate distances. For example, if a car is moving at -40 miles per hour, the absolute value function can be used to find the actual distance traveled, which is 40 miles per hour.
2. Optimization: The absolute value function is also used in optimization problems, such as minimizing or maximizing a function. For instance, in operations research, the absolute value function is often used to model cost, profit, or distance.

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