Relations and Functions Overview

Relations and Functions Overview

• Relations can be represented in various forms, including tables, graphs, and mappings.
• A key aspect of relations is identifying the domain (set of x-values) and range (set of y-values).

Example Relations and Their Domains/Ranges

• Relation: {(4, −2), (−1, 1), (2, −3), (3, 0)}

  • Domain: {-1, 2, 3, 4}
  • Range: {-3, -2, 0, 1}

• Relation: {(3, 4), (1, −2), (4, −1), (2, 2)}

  • Domain: {1, 2, 3, 4}
  • Range: {-2, -1, 2, 4}

• Relation: {(3, −4), (2, 0), (−4, −1), (0, −3)}

  • Domain: {-4, 0, 2, 3}
  • Range: {-4, -3, -1, 0}

• Relation: {(3, 2), (−2, 4), (4, −4), (4, 0), (−1, −3)}

  • Domain: {-2, -1, 3, 4}
  • Range: {-4, -3, 0, 2, 4}

• Relation: {(-1, -5), (2, -3), (3, -2), (5, 1), (-4, 2)}

  • Domain: {-4, -1, 2, 3, 5}
  • Range: {-5, -3, -2, 1, 2}

Function Determination

• To qualify as a function, each x-value must correspond to exactly one y-value.

• Example: { (-1, 2), (2, 51), (1, 3), (8, 22), (9, 51)}

  • Function: Yes
  • Domain: {-1, 2, 1, 8, 9}
  • Range: {2, 51, 3, 22}

• Example: {(1, -2), (-2, 0), (-1, 2), (1, 3)}

  • Function: No (x=1 maps to -2 and 3)
  • Domain: {-2, -1, 1}
  • Range: {-2, 0, 2, 3}

• Example: {(1, 1), (2, 2), (3, 5), (4, 10), (5, 15)}

  • Function: Yes
  • Domain: {1, 2, 3, 4, 5}
  • Range: {1, 2, 5, 10, 15}

Mappings and Functions

• Mappings visually represent relations.

• Example: {(-2, -1), (0, 3), (5, 4), (-2, 3)}

  • Function: No (x=-2 maps to -1 and 3)
  • Domain: {-2, 0, 5}
  • Range: {-1, 3, 4}

• Example: {(-1, 5), (0, 3), (2, 3), (3, -1)}

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

• Example: {(-1, 7), (0, -3), (1, 10), (0, 7)}

  • Function: No (x=0 maps to -3 and 7)
  • Domain: {-1, 0, 1}
  • Range: {-3, 7, 10}

Mapping Diagram

• A mapping diagram illustrates the relationship between elements of the domain and range.
• Example: {(2, 4), (-8, 0), (1, 5), (3, 1)}
  • Function: Yes (each x-value is unique)