General Mathematics Lesson 1: Functions and Relations

Functions and Relations

• A relation is a set of ordered pairs (x, y).
• The domain consists of all x-components (independent variable).
• The range consists of all y-components (dependent variable).

Ordered Pairs Example

• Given ordered pairs {(1,3), (2,4), (5,6), (7,8)}:
  • Domain: {1, 2, 5, 7}
  • Range: {3, 4, 6, 8}
• Given ordered pairs {(-2,1), (0,-1), (5,3), (0,7)}:
  • Domain: {-2, 0, 5}
  • Range: {1, -1, 3, 7}

Definition of a Function

• A function establishes a relationship where each x-value corresponds to exactly one y-value.

Representations of Functions

1. Mapping (one-to-one):

   • Each domain x maps to one unique range y.
   • Example:
     • Domain (x): 1, 3, 5, 7
     • Range (y): 2, 4, 6, 8

2. Mapping (one-to-many):

   • A single domain x can map to multiple y-values.
   • Example:
     • Domain (x): 2, 5, 7
     • Range (y): 3, 4, 6, 8

3. Mapping (many-to-one):

   • Multiple domain x-values can correspond to a single range y-value.
   • Example:
     • Domain (x): 2, 3, 4, 6
     • Range (y): 5, 7, 8

Function Identification

• To determine if a set is a function, a vertical line test can be applied on its graph:
  • A relation is a function if any vertical line intersects the graph at only one point.

Function Examples

• Example of a function: {(1,1), (2,2), (3,3), (4,4)}
• Example of a non-function: {(1,0), (0,1), (-1,0), (0,-1)}

Tables of Values

• Functions can be presented in tabular format.
• Example table with values:
  • x: 1, 2, 3, 4
  • y: 20, 40, 60, 80

Summary

• All functions are relations, but not all relations are functions.
• Understanding the distinction between functions and relations is essential for further mathematical concepts.