Basic Introduction to Domain and Range

Understanding Domain and Range

• Domain: The set of all possible x-values (inputs) for the function.
• Range: The set of all possible y-values (outputs) for the function.

Interval Notation

• Used to express the domain and range.
• Brackets [ ]: Include the endpoint.
• Parentheses ( ): Do not include the endpoint.
• Infinity (∞): Always uses parentheses, as infinity is not a concrete number.

Examples

Example 1: Simple Continuous Function

• Domain: Lowest x-value = -4, Highest x-value = 3
  • Interval Notation: [-4, 3]
• Range: Lowest y-value = -5, Highest y-value = 4
  • Interval Notation: [-5, 4]

Example 2: Graph with Open and Closed Circles

• Domain: Lowest x-value = -6 (not included), Highest x-value = 6 (included)
  • Interval Notation: (-6, 6]
• Range: Lowest y-value = -4 (not included), Highest y-value = 5 (included)
  • Interval Notation: (-4, 5]

Example 3: Graph with Arrows (Extending to Infinity)

• Domain: Lowest x-value = 1 (included), extends to infinity
  • Interval Notation: [1, ∞)
• Range: Lowest y-value = 2 (included), extends to infinity
  • Interval Notation: 2, ∞)

Example 4: Downward Parabola

• Domain: From negative infinity to positive infinity
  • Interval Notation: (-∞, ∞)
• Range: Lowest y-value = -∞, Highest y-value = 3 (included)
  • Interval Notation: (-∞, 3]

Example 5: Graph with Jumps

• Domain: Lowest x-value = -6 (not included), x can be negative one (included), Highest x-value = 5
  • Interval Notation: (-6, -1] ∪ [-1, 5]
• Range: Lowest y-value = -4 (not included), Highest y-value = 4 (included) with a gap between -2 and 1
  • Interval Notation: (-4, -2] ∪ [1, 4]

Example 6: Complex Graph with Multiple Sections

• Domain: Covers several disconnected intervals.
  • Interval segments:
    • [-8, -4)
    • [-2, 5)
    • [7, ∞)
  • Interval Notation: [-8, -4) ∪ [-2, 5) ∪ [7, ∞)
• Range: Analyze each part separately and find unions.
  • Includes values from -6 to 2 and 5 to infinity.
  • Interval Notation: (-6, 2] ∪ 5, ∞)

Tips for Determining Domain and Range

• Examine the x-values for domain and y-values for range.
• Notice open and closed circles for inclusion/exclusion of endpoints.
• Identify any arrows that indicate continuation to infinity.
• Use union symbols (∪) to connect disjoint intervals, especially when dealing with gaps or separate parts of the graph.

This summary provides a structured approach to understanding how to determine the domain and range of functions graphically, using interval notation. Practice with multiple examples will enhance understanding.