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Understanding Relations and Functions

May 18, 2025

Relations and Functions

Introduction to Relations

  • A relation is a set of pairs of input and output values.
  • Ordered pairs represent these relations, with "x" as input and "y" as output.
  • The domain refers to the set of all x-values (input values).
  • The range refers to the set of all y-values (output values).

Determining Domain and Range

  • Example 1:
    • Domain: -3, 0, 2 (in ascending order).
    • Range: 1, 4, 5 (already in ascending order).
  • Example 2:
    • Domain: -2, 1, 3.
    • Range: -2, 3, 4, 7.

Identifying Functions

  • A relation is a function if each input value has a unique output value.
  • First Relation:
    • Ordered Pairs: (2, 1), (-3, 4), (0, 5)
    • Each input corresponds to one output.
    • Conclusion: It is a function.
  • Second Relation:
    • Ordered Pairs: (1, 3), (-2, 4), (3, -2), (-2, 7)
    • Input -2 corresponds to two outputs (4 and 7).
    • Conclusion: It is not a function.
  • Quick Check: Look for repeating x-values with different y-values to identify non-functions.

Mapping Diagrams

  • Visual representation of a relation.
  • Shows domain on one side, range on the other.
  • Mapping Example:
    • First Mapping: Yes, it is a function.
    • Second Mapping: No, it is not a function due to repeating x-values.

Function Tables

  • Lists input values (x) and corresponding output values (y).
  • Identifies if a relation is a function based on repeating x-values.
  • Example:
    • Two identical x-values with different y-values indicate a non-function.

Graphs and the Vertical Line Test

  • Vertical Line Test: Determines function status of a graphed relation.
  • If a vertical line touches the graph at more than one point, it is not a function.
  • Examples:
    • Graph with single intersection per line location: Function.
    • Graph with multiple intersections: Not a function.
    • Circle: Not a function as it fails the vertical line test.

Conclusion

  • Relations and functions provide a way to connect input-output values.
  • Function identification relies on unique output for each input and passing the vertical line test for graphs.

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