Lecture Notes on Functions and Relations

Introduction to Relations

• A relation is defined as any set of ordered pairs.
• Domain: Set of all x components of the ordered pairs.
• Range: Set of all y components of the ordered pairs.
• The domain can be thought of as the input values and the range as the output values.

Example of Domain and Range

1. Given Relation: {(1, 3), (2, 4), (5, 7), (6, 8)}

   • Domain: {1, 2, 5, 6}
   • Range: {3, 4, 7, 8}

2. Given Relation: {(−2, 4), (−1, 1), (0, 0), (5, −2)}

   • Domain: {−2, −1, 0, 5}
   • Range: {4, 1, 0, −2}

Definition of Functions

• A function is a special type of relation where each member of the domain is paired with exactly one member of the range.
• For a relation to be a function, no two ordered pairs can have the same x value with different y values.

Examples of Functions

1. Function Example 1: {(1, 2), (2, 3), (3, 4), (4, 5)}

   • Each x-value has a unique y-value.

2. Function Example 2: {(1, 1), (2, 2), (3, 3), (4, 4)}

   • Each x-value corresponds to the same y-value (identity function).

3. Not a Function Example: {(1, 0), (0, 1), (−1, 0), (0, −1)}

   • The x-value 0 corresponds to two different y-values (1 and −1).

4. Function Example 3: {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}

   • Each x-value maps to a unique y-value.

Mapping Diagrams

• Functions can also be represented through mapping diagrams where elements of the domain are connected to elements of the range using arrows.
• Example Mapping Test:
  • If each x component corresponds to a unique y component, it is a function.

Vertical Line Test for Functions

• A graph represents a function if any vertical line drawn through it touches the graph at exactly one point.
• Examples:
  • Straight lines and certain curves (like parabolas) may represent functions.
  • Ellipses and hyperbolas do not represent functions as they fail the vertical line test.

Conclusion

• Understanding the concepts of relations and functions is crucial for higher-level mathematics.
• For any questions or clarifications, comments can be left for discussion.