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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.
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