Understanding Functions and Relations

Lecture Notes: Introduction to Functions and Relations

General Introduction

• Audience: Incoming Grade 11 students
• Focus: Overview of general mathematics, specifically functions
• Key Concepts: Domain, Range, Ordered Pairs, Functions, Relations

Key Definitions

• Domain: Set of the first coordinates in ordered pairs.
• Range: Set of the second coordinates in ordered pairs.
• Relation: A set of ordered pairs
• Function: A relation where each element of the domain is associated with exactly one element of the range.

Identifying Domains and Ranges

• Example of ordered pairs:
  • (1, -1), (2, -3), (0, 5), etc.
• Domain: Extract x-values (e.g., -1, 0, 1, 2, 4)
• Range: Extract y-values (e.g., -5, -4, -3, -1, 3, 5)

Representation of Functions

Using Ordered Pairs

• Examine if x-values repeat for different y-values if yes, not a function.
• Example Relations:
  • Relation F: Function (Example: x-values 1, 2, 3, 4 do not repeat)
  • Relation G: Not a function (Example: x-value 1 is repeated)
  • Relation H: Function

Using Mapping Diagrams

• Input and Output: Like a machine processing inputs into outputs
• Function if each input corresponds to one output.
  • Example 1: Inputs 10, 20, 30 with unique outputs is a function.
  • Example 2: Multiple inputs to one output is not a function.

Using Graphs

• Vertical Line Test: Determines if graph represents a function
  • If any vertical line intersects graph more than once, it’s not a function.
  • Examples:
    • Certain linear graphs pass the test, thus are functions.
    • Circular graphs fail the test, thus are not functions.

Using Equations

• Types of equations that are functions:
  • Linear (e.g., y = 2x + 1) and Quadratic (e.g., y = x² - 2x + 2)
• Failure cases:
  • Equations like x² + y² = 1 are not functions (circle, fails vertical line test)

Practice Questions

1. Determine Functions from Ordered Pairs
   • Example Sets: Identify if they are functions based on repeated x-values.
2. Mapping Diagram Evaluations
   • Analyze inputs and outputs.
3. Graph Analysis
   • Apply vertical line test.
4. Equation Analysis
   • Identify if given equations represent functions.

Conclusion

• Importance of understanding different representations and identifications of functions.
• Encouragement to keep practicing identifying functions through various representations.
• Preparation for further discussions on domain and range in future lessons.

References

• Books and educational materials used in the lecture.
• Additional resources for self-study and practice.