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

Absolute Value Function

The absolute value function, denoted as |x|, is a mathematical function that gives the absolute value or magnitude of a real number

The absolute value function, denoted as |x|, is a mathematical function that gives the absolute value or magnitude of a real number. It is defined as follows:

For a given real number x,
– If x is positive or zero, |x| = x.
– If x is negative, |x| = -x.

In simpler terms, the absolute value of a number is the distance of that number from zero on the number line, always resulting in a non-negative value.

Here are some key properties and characteristics of the absolute value function:

1. Symmetry: The absolute value function is symmetric about the y-axis. This means that |x| = |-x| for all real numbers x.

2. Non-negativity: The absolute value is always non-negative. |x| ≥ 0 for all real numbers x. If x = 0, then |x| = 0.

3. Distance interpretation: The absolute value of a number represents the distance between that number and zero on the number line. For example, |5| = 5 means that 5 is 5 units away from zero, and |-3| = 3 means that -3 is 3 units away from zero.

4. Piece-wise definition: The absolute value function can be defined using a piece-wise function notation:
|x| =
x for x ≥ 0,
-x for x < 0. 5. Graph: The graph of the absolute value function is a "V" shape, with its vertex at the origin (0, 0). It consists of two linear segments, one increasing for positive x-values and the other decreasing for negative x-values. 6. Absolute value inequalities: Absolute values can be used in solving and graphing inequalities. For example, |x - 3| > 4 represents all the x-values that are more than 4 units away from 3 on the number line.

When working with the absolute value function, it is important to understand these properties and use them to solve equations, evaluate expressions, and graph functions effectively.

More Answers: