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Understanding Relations and Functions

May 2, 2025

Relations and Functions Lecture Notes

Definition of a Relation

  • A relation is a set of pairs of input and output values.
  • Represented by ordered pairs ((x, y)):
    • x-value: Input value, associated with the domain.
    • y-value: Output value, associated with the range.

Domain and Range

Example 1

  • Domain (x-values) in Ascending Order:
    • (-3, 0, 2)
  • Range (y-values) in Ascending Order:
    • (1, 4, 5)

Example 2

  • Domain (x-values):
    • (-2, 1, 3)
  • Range (y-values):
    • (-2, 3, 4, 7)

Function Criteria

  • Definition of a Function: Every input value must have only one output value.
  • If an input corresponds to multiple outputs, it is not a function.

Example Analysis

  • First Relation:

    • Ordered pairs: ((2, 1), (-3, 4), (0, 5))
    • Each input value corresponds to only one output value (\Rightarrow) This is a function.

  • Second Relation:

    • Ordered pairs: ((1, 3), (-2, 4), (3, -2), (-2, 7))
    • Input (-2) corresponds to multiple outputs (4) and (7) (\Rightarrow) Not a function.
    • Quick recognition: Repeating x-values with different y-values indicate it is not a function.

Mapping Diagrams

  • First Relation:

    • Domain: (-2, 1, 3)
    • Range: (-6, 0, 4)
    • Function since every input has a single output.

  • Second Relation:

    • Domain: (-2, 0, 3)
    • Range: (-1, 1, 2, 5)
    • Not a function since input (0) corresponds to multiple outputs (-1) and (5).

Function Table

  • Input Values (Domain): (-3, 1, 1, 3, 5)
  • Output Values (Range): Matching pairs based on relation
  • Repeated x-values with different y-values indicate non-function.

Vertical Line Test

  • Purpose: Determine if a graph represents a function.
  • Method:
    • Draw vertical lines across the graph.
    • If a vertical line touches the graph at more than one point, it is not a function.

Graph Analysis

  1. First Graph:
    • Vertical line touches only once (\Rightarrow) Function.
  2. Second Graph:
    • Multiple intersections (\Rightarrow) Not a function.
  3. Third Graph:
    • Single intersection per vertical line (\Rightarrow) Function.
  4. Circle Graph:
    • Two points of intersection (\Rightarrow) Not a function.

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