Funciones de uso frecuente (Parte 2): Función Valor Absoluto

Understanding the Absolute Value Function

Introduction to Absolute Value

• The discussion begins with an introduction to the absolute value function, emphasizing its frequent use in mathematical studies.

Definition of Absolute Value

• The absolute value function is represented by enclosing the variable x within vertical bars, denoted as |x| .

• It is defined such that:

• If x geq 0 , then |x| = x . For example, |7| = 7 .

• If x < 0 , then |x| = -x . For instance, |-3| = -(-3) = 3 .

Geometric Interpretation

• Geometrically, the absolute value represents the distance from a number to zero on the number line.

• This means that all results of absolute values are non-negative; for example:

• |7| = 7

• |-3| = 3

• |0| = 0

Constructing a Table of Values

• A table is constructed using various real numbers:

• For negative values:

• |-3| = 3

• |-2| = 2

• |-1| = 1

• For zero and positive values:

• |0| = 0

• |1| = 1

• |2| = 2

• |3| = 3

Graphing the Absolute Value Function

• Points from the table are plotted on a Cartesian plane:

• Points include (-3,3), (-2,2), (-1,1), (0,0), (1,1), (2,2), and (3,3).

• Connecting these points forms the graph of the absolute value function.

Domain and Range Analysis

• Using a shadow projection technique:

• The domain includes all real numbers ( x ∈ ℝ).

• The range consists of non-negative values ( y ≥ 0), which can be expressed as [0, ∞).

Behavior of the Function

• The function exhibits different behaviors based on intervals:

• Decreasing for values where x < 0.

• Increasing for values where x > 0.