Lecture Notes: Upgrading Functions

Introduction

• Started with the concept of relationships.
• Upgraded to functions.
• Today's focus: Upgrade functions to better functions.

Function vs. Relationship

• Functions: Written as f(x) or of x notation.
• Definition: Every x has only one y.
• Example: Points (1,2), (2,2), (3,2) form a function.
  • Passes vertical line test.

One-to-One Functions

• Definition: Every y also has only one x.
• Characteristics:
  • More restrictive than regular functions.
  • Treat functions as operators (like addition/subtraction).

Hierarchy of Relationships

• Relationship: Needs an x and a y.
• Function: Every x has only one y.
• One-to-One Function: Every y has only one x.
  • Most restrictive and useful.
  • Can be inverted (talked about in inverse functions).

Examples of Functions and Relationships

• Equation 1: y = (x-2)^2 + 2 (Parabola)
  • Passes vertical line test; a function.
  • Fails horizontal line test; not one-to-one.
• Equation 2: x^2 + y^2 = 16 (Circle)
  • Fails vertical line test; not a function.

Testing Functions

• Vertical Line Test: For checking functions.
• Horizontal Line Test: For checking one-to-one functions.

Algebraic Analysis

• Example 1: y = (x-2)^2 + 2
  • Solve for x: y = 5 gives two solutions due to square root.
  • Indicates not one-to-one.
• Example 2: x^2 + y^2 = 16
  • Solve for y: y = ±√(16 - x^2)
  • Two solutions indicate it's a relationship.

Square Roots and Functions

• Square Roots:
  • Introduce two solutions (± sign).
  • Challenge in maintaining function status.
• Odd Roots:
  • Do not require ± sign.
  • Maintain function status.

Conclusion

• Upgrading functions involves adding more restrictions.
• Understanding relationship, functions, and one-to-one functions is crucial for mathematical analysis and operations.
• Beware of squares and square roots as they can complicate function status.