Functions Overview

Introduction

• Functions are introduced in year 11.
• Important for understanding concepts in year 12.

What is a Function?

• Relation: A set of ordered pairs with coordinates (x, y) related by a rule.
  • Example: x² + y² = 25 (equation of a circle).
• Function: A specific type of relation.
  • Each x-value has one unique y-value.
  • Example: y = x² (parabola), cubic functions, and non-vertical straight lines.

Differentiating Relations and Functions

• Circle Example: An x-coordinate can correspond to two y-values, thus not a function.
• Parabola Example: Each x-value corresponds to one y-value, making it a function.
• Two x-values can have the same y-value, but a single x-value can't have more than one y-coordinate.

Vertical Line Test

• Used to determine if a graph is a function.
• Draw a vertical line through the graph; if it intersects more than once, it's not a function.

Function Notation

• If y depends on x, written as y = f(x).
• f(x) indicates x being substituted into the function.
• Example: f(1) for y = x² results in 1² = 1.

Domain and Range

• Domain: The set of x-values for which a function is defined.
• Range: The set of y-values for which a function is defined.
• Example: y = x² has domain of all real x-values and range of y ≥ 0.

Specific Function Types

Hyperbolas

• Definition: Discontinuous functions in the form xy = k or y = k/x.
• Asymptotes on x and y axes.
• If k > 0, curves in 1st and 3rd quadrants; if k < 0, 2nd and 4th.

Circles

• Equation: x² + y² = r² or (x-a)² + (y-b)² = r².
• Center (a, b), radius r.
• Domain and range related to radius.

Semicircles

• Equations: y = √(r² - x²) or y = -√(r² - x²).
• Domain and range depend on radius and position relative to x-axis.

Absolute Values

• Represent positive distance from zero: |x|.
• Graph is V-shaped, symmetrical about the y-axis.

Even and Odd Functions

• Even Functions: Symmetrical about the y-axis.
  • Condition: f(x) = f(-x).
• Odd Functions: Symmetrical about the origin.
  • Condition: f(x) = -f(-x).

Practice Examples

• Substitution examples using function notation and solving.
• Proofs of functions being even, odd, or neither.

Conclusion

• Understanding functions includes recognizing their types, notations, and properties.
• Graph interpretation and domain/range identification are crucial skills.