Lecture Notes: Relations and Functions

Introduction

• Relations and Functions
  • A relation is a set of ordered pairs.
  • The domain is the set of all x-components (inputs).
  • The range is the set of all y-components (outputs).
  • A function is a type of relation where each element of the domain is paired with exactly one element of the range.

Domain and Range

• Example 1: Relation - (1,3), (2,4), (5,7), (6,8)
  • Domain: 1, 2, 5, 6
  • Range: 3, 4, 7, 8
• Example 2: Relation - (-2,4), (-1,1), (0,0), (1,5), (2,-2)
  • Domain: -2, -1, 0, 1, 2
  • Range: 4, 1, 0, 5, -2

Identifying Functions

• A relation is a function if no two ordered pairs have the same x-value but different y-values.
• Examples:
  1. (1,2), (2,3), (3,4), (4,5) - Function
  2. (1,1), (2,2), (3,3), (4,4) - Function
  3. (1,0), (0,1), (-1,0), (0,-1) - Not a Function (0 maps to two different y-values)
  4. (-2,4), (-1,1), (0,0), (1,1), (2,4) - Function

Mapping Diagrams

• Functions can be represented using mapping diagrams.
• Example Mapping Diagrams:
  • Diagram 1: 1->3, 2->5, 3->9, 4->7, 5->33 - Function
  • Diagram 2: 7->1, 8->0, 9->0 - Not a Function (multiple inputs map to the same output)
  • Diagram 3: (11, 13, 17, 19, 23) - Not a Function (inputs do not map uniquely)

Graphical Representation

• Functions can be identified using the Vertical Line Test.
  • A graph is a function if any vertical line drawn through the graph touches it at exactly one point.
• Graph Tests:
  • Graph A: Passes the vertical line test - Represents a Function
  • Graph B: Straight line - Represents a Function
  • Elliptic Graph: Fails the vertical line test - Not a Function
  • Hyperbolic Graph: Not a Function

Conclusion

• Understanding the concepts of relations and functions is critical in mathematics.
• Using mapping diagrams and the vertical line test help identify functions.

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