Algebra Basics: Functions

Introduction

• Presenter: Rob from Math Antics
• Topic: Functions in algebra
• General definition: Outside math, functions describe what something does.
• Math definition: A function connects one set to another in a specific way.

Understanding Sets

• Definition: A collection of things (numbers, letters, names, etc.).
• Notation: Elements in curly brackets, e.g., {1, 2, 3}.
• Types:
  • Finite sets (e.g., letters of the alphabet)
  • Infinite sets (e.g., all integers)

Functions Explained

• Input and Output Sets:
  • Input set: The Domain
  • Output set: The Range
• Function Table: Lists input and output values side-by-side.
• Example with Polygons:
  • Input: {triangle, square, pentagon, hexagon, octagon}
  • Function Rule: Output the number of sides.
  • Output: {3, 4, 5, 6, 8}

Algebraic Functions

• Example Equation: y = 2x
• Domain: Set of x values
• Range: Set of y values
• Function Table Example: (x, y pairs)
  • (1, 2)
  • (2, 4)
  • (3, 6)

Function Limitations

• One-to-Many Relations: Not allowed in functions.
• Example: y^2 = x
  • Input: x = 4
  • Outputs: y = 2 and y = -2
  • Conclusion: Not a function.
• Rule: Each input has exactly one output.

More Examples

• Linear Function: y = x + 1
  • Function Table: (x, y pairs)
    • (-3, -2)
    • (-2, -1)
    • (-1, 0)
    • (0, 1)
    • (1, 2)
    • (2, 3)
    • (3, 4)
• Graphing Functions: Plot ordered pairs on the coordinate plane.
  • Example Graph: Straight line (linear function)
  • Passing the Vertical Line Test confirms it is a function.

Vertical Line Test

• Purpose: To check if a graph represents a function.
• Method: Move a vertical line across the graph.
  • If it intersects the graph at only one point for each x value, it is a function.
  • Example of failure: y^2 = x

Function Notation

• Standard Equations: y = 2x and y = x + 1
• Function Notation: f(x) = y
• Explanation:
  • f is the name of the function
  • f(x) means the function of x
• Evaluation Example:
  • Function: f(x) = 3x + 2
  • Evaluate for x = 4: f(4) = 3(4) + 2 = 14

Summary

• Definition: Functions relate inputs to exactly one output.
• Domain: Set of all inputs.
• Range: Set of all outputs.
• Graphing: Inputs and outputs can be graphed as ordered pairs.
• Practice: Important to practice using function concepts.

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