Understanding Functions and Their Properties

Sep 8, 2024

Lecture on Functions

Definition of a Function

  • A function is an ordered pair where each input (x) has exactly one output (y).
  • Every x can only have one corresponding y value.

Examples of Functions

  • Example 1:
    • Inputs: x, y, z
    • Outputs: a, b, c
    • Each input has a single distinct output.
  • Example 2:
    • Input: City
    • Output: Area Code
    • Grand Rapids maps to 616, Spring Lake also maps to 616, Muskegon maps to 231.
    • Different inputs can share the same output, but each input has one specific output.

Example of Not a Function

  • In a table where inputs and outputs are listed:
    • If an input has two different outputs, it's not a function.
    • Example: Input -2 maps to outputs -1 and 2.
    • An input with more than one output indicates it's not a function.

Determining Functions from Tables

  • In a table:
    • If every x has only one corresponding y, it is a function.

Vertical Line Test

  • A method to determine if a graph represents a function.
  • Vertical Line Test:
    • Equation for a vertical line is x = a, meaning all x's are the same.
    • Draw a vertical line through the graph:
      • If it intersects the graph at only one point, every x has only one y, indicating a function.
      • If it intersects at more than one point, the graph does not represent a function.
  • Example:
    • A vertical line through multiple y values for the same x value means it's not a function.

Key Takeaway

  • Use the vertical line test to easily determine if a graph can be a function:
    • If any vertical line crosses the graph more than once, it's not a function.