Lecture on Functions and Function Notations

Introduction to Functions

• Functions are built upon relationships.
• A mathematical relationship is defined by a single point, represented as (x, y).
• Example: Point (3, -1) indicates a relationship between x = 3 and y = -1.
• Equations represent relationships as infinite points.

Vocabulary of Relationships

• Relationships can be represented as a collection of points within curly braces.
• Domain: Set of all x-values in a relationship.
• Range (also referred to as Image): Set of all y-values in a relationship.
  • Term "Image" is less commonly used today.

Understanding Domain and Range

• Domain: Easy to determine, represents potential inputs (x-values).
• Range: Can be more complex, dependent on the equation's depiction.

Example

• Equation: y = 3x + 1
  • Domain: All real numbers, expressed as (-∞, ∞) or ℝ.
  • Range: Also all real numbers for a linear equation.

Domain Issues

• Square Roots:

  • Issue arises when x under a square root results in a negative number.
  • Example: √(x + 2)
    • Domain: x + 2 ≥ 0, or x ≥ -2.
    • Domain notation: [-2, ∞).

• Division by Zero:

  • Issue arises when x in the denominator makes the denominator zero.
  • Example: 3/(x + 1)
    • Domain: Exclude x = -1.
    • Domain notation: (-∞, -1) ∪ (-1, ∞).

Concepts of Inclusion and Exclusion

• Inclusion: Uses square brackets, number is included in the set.
• Exclusion: Uses parentheses, number is not included.
• Infinity: Always excluded as it represents a direction, not a number.

Summary

• Focus on understanding domains because they define what x-values can be used without causing errors.
• Relationship and domain concepts are foundational for understanding functions.
• Functions will be explored in the following discussion.