Overview
This lecture explains the difference between a relation and a function, using ordered pairs and their mappings, and evaluates whether a given relation meets the definition of a function.
Relations and Functions
- A relation is a set of ordered pairs that associates elements from the domain (inputs) to the range (outputs).
- The domain is the set of all possible input values for a relation.
- The range is the set of all possible output values associated with the domain by the relation.
- A function is a special type of relation in which each input in the domain is associated with exactly one output in the range.
- In a function, each domain value has only one mapping; there is no ambiguity in what it maps to.
Examples of Relations and Functions
- If 1 maps to 2, 2 maps to 2, and 3 maps to -7, this relation is a function because each domain value maps to one range value.
- If 1 maps to 2, 2 maps to -3, and 1 also maps to 4, this relation is not a function because 1 maps to more than one value.
Determining Function from a Set of Ordered Pairs
- To check if a relation is a function, examine if any input maps to more than one output.
- Example relation with domain: -3, -2, 0, 3 and range: 2, 4, 5, 6, 8.
- Ordered pairs: (-3,2), (-2,4), (0,5), (-2,6), (3,8).
- The input -2 maps to both 4 and 6, so the relation is not a function.
Key Terms & Definitions
- Relation — A set of ordered pairs associating elements from a domain to a range.
- Domain — The set of all possible input values for a relation.
- Range — The set of all possible output values for a relation.
- Function — A relation where each input in the domain maps to exactly one output in the range.
Action Items / Next Steps
- Practice determining if various relations are functions by checking if any input maps to more than one output.