Relations vs Functions
Understanding mathematical relationships
What is a Relation?
A relation is any set of ordered pairs \((x, y)\) that shows a connection between elements of two sets. The set of all first elements (x-values) is called the domain, and the set of all second elements (y-values) is called the range.
What is a Function?
A function is a special type of relation where each input (x-value) has exactly one output (y-value). In other words, no x-value can be paired with more than one y-value.
Key Rule: Every function is a relation, but not every relation is a function!
Comparing Relations and Functions
Relation
Definition: Any pairing between sets
Rule: An x-value CAN map to multiple y-values
Example:
\(\{(1, 2), (1, 3), (2, 4)\}\)
Here, 1 maps to both 2 and 3
Function
Definition: A special relation
Rule: Each x-value maps to exactly ONE y-value
Example:
\(\{(1, 2), (2, 4), (3, 6)\}\)
Each x has only one y
Types of Mappings
One-to-One (Function)
Each element in the domain maps to a unique element in the range.
Example: \(f(x) = 2x\)
Many-to-One (Function)
Multiple elements in the domain can map to the same element in the range.
Example: \(f(x) = x^2\) (both -2 and 2 give 4)
One-to-Many (NOT a Function)
One element in the domain maps to multiple elements in the range.
Example: \(y^2 = x\) (when x=4, y can be 2 or -2)
The Vertical Line Test
How It Works
To determine if a graph represents a function, draw vertical lines across the graph:
- If every vertical line crosses the graph at most once → It IS a function
- If any vertical line crosses the graph more than once → It is NOT a function
Interactive Mapping Diagram
Create Your Own Mapping
Click elements to create mappings and see if your relation is a function
Domain (X)
Range (Y)
Create mappings by clicking a domain element, then a range element.
Representing Functions
Ways to Represent a Function
1. Set of Ordered Pairs: \(\{(1,2), (2,4), (3,6)\}\)
2. Mapping Diagram: Arrows from domain to range
3. Table of Values:
| x | 1 | 2 | 3 |
|---|---|---|---|
| y | 2 | 4 | 6 |
4. Equation: \(y = 2x\) or \(f(x) = 2x\)
5. Graph: A visual representation on the coordinate plane
Practice Problems
Question 1: Is the relation \(\{(2,3), (4,5), (2,7), (6,9)\}\) a function?
Show Solution
No, this is NOT a function.
The x-value 2 is paired with both 3 and 7.
In a function, each x-value can only have one y-value.
Question 2: Determine whether \(y = x^2 + 1\) is a function.
Show Solution
Yes, this IS a function.
For any value of x, there is exactly one value of y.
Example: When x = 3, y = 9 + 1 = 10 (only one answer)
Question 3: Is the equation \(x = y^2\) a function of x?
Show Solution
No, this is NOT a function (of y in terms of x).
When x = 4, y could be 2 or -2.
Each x-value would have two possible y-values.
Question 4: Given the mapping: A→1, B→2, C→2, D→3. Is this a function?
Show Solution
Yes, this IS a function.
Each element in the domain (A, B, C, D) maps to exactly one element in the range.
It's okay for B and C to both map to 2 (many-to-one is allowed).
Key Takeaways for CSEC
- A function assigns exactly ONE output to each input
- Use the vertical line test on graphs
- Check ordered pairs for repeated x-values with different y-values
- Many-to-one IS a function; one-to-many is NOT
- Domain = set of inputs; Range = set of outputs