Lecture Notes on Relations and Functions

Overview of Relations

• Definition: A relation is a set of pairs of input (x) and output (y) values.
• Ordered Pairs: Each pair consists of an x-value (input) and a y-value (output).
• Domain and Range:
  • Domain: Set of all x-values (input values).
  • Range: Set of all y-values (output values).

Example Relations

First Relation:

• Ordered Pairs: (-3, 4), (0, 5), (2, 1)
• Domain: -3, 0, 2
• Range: 1, 4, 5

Second Relation:

• Ordered Pairs: (-2, 4), (1, 3), (-2, 7), (3, -2)
• Domain: -2, 1, 3
• Range: -2, 3, 4, 7

Determining Functions

• Definition of a Function:
  • For a relation to be a function, every input (x-value) must correspond to one and only one output (y-value).
  • If an input corresponds to multiple outputs, it is not a function.
• First Relation:
  • Each x-value corresponds to one y-value. It is a function.
• Second Relation:
  • The input -2 corresponds to both 4 and 7. It is not a function.

Identifying Functions via Repeating X-Values

• If there are repeating x-values corresponding to different y-values, the relation is not a function.

Mapping Diagrams

Mapping for First Relation:

• Domain: -2, 1, 3
• Range: -6, 0, 4
• Mapping:
  • -2 → 0
  • 1 → 4
  • 3 → -6
  • It is a function.

Mapping for Second Relation:

• Domain: -2, 0, 3
• Range: -1, 1, 2, 5
• Mapping:
  • -2 → 1
  • 0 → 5
  • 3 → 2
  • 0 → -1
  • It is not a function.

Function Tables

• Definition: List input values (x-values) next to output values (y-values).
• Example Table:
  • Inputs: -3, 1, 1, 3, 5
  • Corresponding Outputs:
    • -3 → 2
    • 1 → 1 or 4
    • 3 → 7
    • 5 → -4
  • This relation is not a function.

Vertical Line Test

• Definition: A graphical method to determine if a relation is a function.
• Application:
  • Draw a vertical line through the graph:
    • If the line touches the graph at more than one point, it is not a function.
• Examples:
  • A single intersection point implies it is a function.
  • Multiple intersection points imply it is not a function.

Conclusion

• Understanding relations and functions is essential in mathematics.
• Use methods like the mapping diagram and vertical line test to determine functions.