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.