General Mathematics Lesson 1: Functions and Relations

Functions and Relations

• Definitions
  • 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}

• Another set of ordered pairs: {(-2, 1), (0, -1), (5, 3), (0, 7)}

  • Domain: {-2, 0, 5}
  • Range: {1, -1, 3, 7}

Function Characteristics

• A function maps each element of set x to exactly one element in set y.

Function Representations

• Mapping (One-to-One)

  • Each domain value corresponds to one unique range value.

• Mapping (One-to-Many)

  • A single domain value corresponds to multiple range values.

• Mapping (Many-to-One)

  • Multiple domain values correspond to a single range value.

Function Validation

• Function Test: Ordered pairs

  • {(1, 1), (2, 2), (3, 3), (4, 4)} is a function.
  • {(1, 0), (0, 1), (-1, 0), (0, -1)} is not a function (0 cannot map to both 1 and -1).

• Table of values validation

  • For table with x: -1, 0, 1, 2 and y: 2, -2, 3, 3 - not a function due to the value 3 being repeated for different x-values.

Graphical Representation

• Vertical Line Test:
  • A relation is a function if a vertical line intersects the graph at only one point.
  • If a vertical line intersects at more than one point, it is not a function.

Important Relationships

• All functions are relations, but not all relations are functions.