Overview
This lecture introduces the concept of relations and functions, focusing on identifying when a relation is a function and key differences between the two.
Relations and Their Representation
- A relation pairs elements from one set (domain) with elements from another set (range).
- Relations can be represented using tables, graphs, mappings, or ordered pairs.
- The domain is the set of all possible input values; the range is the set of possible output values.
Definition of a Function
- A function is a special type of relation where each input (domain) has exactly one corresponding output (range).
- Not every relation is a function, but every function is a relation.
- Functions can be represented as equations, tables, mappings, or graphs.
Identifying Functions
- Use the "vertical line test" on a graph: if any vertical line crosses the graph more than once, it's not a function.
- In a table or mapping, a function cannot assign two different outputs to the same input.
- Each element of the domain must pair with only one range element for a relation to be a function.
Examples and Non-Examples
- Example: The relation {(1,2), (2,3), (3,4)} is a function because each input matches with only one output.
- Non-example: The relation {(1,2), (1,3), (2,4)} is not a function because input 1 pairs with two outputs.
Key Terms & Definitions
- Relation — A pairing of elements from one set (domain) with elements from another set (range).
- Function — A relation where each domain element has exactly one range element.
- Domain — The set of all possible input values in a relation or function.
- Range — The set of all possible output values in a relation or function.
- Vertical Line Test — A method to determine if a graph represents a function by checking for multiple outputs per input.
Action Items / Next Steps
- Review textbook examples of relations and functions.
- Practice using tables, mappings, and graphs to identify functions.
- Complete assigned homework problems on distinguishing functions from general relations.