Relations and Functions

Definition of a Relation

• A relation is a set of pairs of input and output values.
• Example: Three ordered pairs where x (input) is associated with the domain and y (output) is associated with the range.

Domain and Range

• Domain: The set of all x values (input values).
• Range: The set of all y values (output values).

Example 1: List Domain and Range

First Relation

• Domain: -3, 0, 2 (in ascending order)
• Range: 1, 4, 5

Second Relation

• Domain: -2, 1, 3
• Range: -2, 3, 4, 7

Determining if a Relation is a Function

• A relation is a function if every input value has exactly one output value.
• Non-function: If an input value has multiple output values.

Checking Relations

First Relation

• Ordered pairs: (2, 1), (-3, 4), (0, 5)
• Each input has one output -> This is a function.

Second Relation

• Ordered pairs: (1, 3), (-2, 4), (3, -2), (-2, 7)
• Input -2 corresponds to both 4 and 7 -> This is not a function.
• Tip: Repeating x values with different y values indicate a non-function.

Mapping Diagrams

• First Relation Mapping:
  • Domain: -2, 1, 3
  • Range: -6, 0, 4
  • Correspondences: -2 -> 0, 1 -> 4, 3 -> -6
  • This is a function.
• Second Relation Mapping:
  • Domain: -2, 0, 3
  • Range: -1, 1, 2, 5
  • Correspondences: -2 -> 1, 0 -> 5, 3 -> 2, 0 -> -1
  • Repeating x values (0 -> 5 and 0 -> -1) -> Not a function.

Function Tables

• Constructing function tables helps visualize x (input) and y (output) relationships.
• Example Table:
  • Inputs: -3, 1, 1, 3, 5
  • Outputs: 2, 2, 4, 7, -4
  • Repeats in x with different y values -> Not a function.

Vertical Line Test (Graph-Based Function Test)

• If a vertical line intersects the graph at more than one point, it is not a function.

Applying Vertical Line Test

Graph Examples

• First Graph: Vertical line touches only at one point -> Yes, it is a function.
• Second Graph: Vertical line touches at multiple points -> No, not a function.
• Third Graph (Circle): Vertical line touches at two points -> No, not a function.
• For graphs where vertical lines touch at only one point per location -> They represent functions.

Summary

• Domain and range list all possible inputs and outputs.
• Repeated x values with different y outputs indicate non-functions.
• Use mapping diagrams and function tables to visualize relationships.
• Vertical line test is a reliable graphical method to determine functions.