Lecture Notes: Functions, Limits, and Continuity

Introduction

• This is the first part of a series (3-4 parts) on Functions, Limits, and Continuity.
• Key terms: Functions, Limits, Continuity.

Functions

Definition

• A function assigns a rule that describes how one quantity depends on another.
• Example: The equation of a line: y = mx + b.
  • x is the input variable (domain).
  • y is the output variable (range).

Ordered Pairs

• A function is defined as a set of ordered pairs (x, y) where no two distinct pairs have the same first number (x).
  • Domain: Set of all possible x values.
  • Range: Set of all possible y values.
• Functions are denoted by small letters: f, g, h.

Function Examples

• For f(x) = x - 1:
  • Example values:
    • x = -3 → y = -7
    • x = 0 → y = -1
    • Range: {-7, -5, -3, -1, 1, 3, 5}
• The relation x² + y² = 9 is not a function as it fails the vertical line test.

Vertical Line Test

• To determine if a graph is a function:
  • Draw vertical lines through the graph.
  • If it intersects more than once, it is not a function.

Mapping and Correspondence

• A function has a one-to-one correspondence: each x maps to exactly one y.
• Example of non-function: x² + y² = 9.
• Example of function: f(x) = x - 1.

Evaluating Functions

• Evaluate functions by straightforward substitution.
• Example for f(x) = x² - 2x + 3:
  • f(2) = 3
  • f(½) = 9/4
  • f(-1) = 6

Domain and Range

Definition

• Domain: All admissible values of x.
• Range: All permissible values of y.

Examples of Finding Domain and Range

• For f(x) = 2x + 3:
  • Domain: All real numbers (−∞, ∞)
  • Range: All real numbers (−∞, ∞)
• For g(x) = √x:
  • Domain: [0, ∞)
  • Range: [0, ∞)
• For h(x) = 2/x:
  • Domain: (-∞, 0) ∪ (0, ∞)
  • Range: (-∞, 0) ∪ (0, ∞)

Asymptotes

Vertical Asymptotes

• Occur where the function approaches infinity as x approaches a certain value.

Horizontal Asymptotes

• Behavior of the function as x approaches positive or negative infinity.

Graphing Functions

Basic Steps

1. Identify the domain and range.
2. Choose suitable x values from the domain and solve for y.
3. Determine behavior of x and y.
4. Plot points on the plane.
5. Smoothly trace the curve.

Example: Sketching Graphs

• For linear functions like f(x) = 2x + 3, use the rise/run technique.
• For parabolas or circles, consider their special properties and transformations.

Conclusion

• Functions are foundational in calculus. Understanding their properties, including limits and continuity, is essential for further studies.
• Next lecture will cover operations on functions and different types of functions.