Relations
A relation is any connection between two sets of values, shown as ordered pairs (x, y).
Examples:
- Students and their heights
- {(1, 3), (2, 5), (2, 7), (4, 1)}
Functions
A function is a special relation: every input has exactly one output.
| Relation | Function? | Why |
|---|---|---|
| {(1, 2), (2, 4), (3, 6)} | Yes | each x has one y |
| {(1, 2), (1, 5), (3, 6)} | No | x = 1 has two outputs |
| {(1, 4), (2, 4), (3, 4)} | Yes | outputs can repeat; inputs can’t |
Think of a vending machine: pressing B4 must always give the same snack. If B4 sometimes gave chips and sometimes cookies, it wouldn’t be a function.
Ways to represent relations
Mapping diagram: arrows from each input to its output(s). If any input has two arrows, it’s not a function.
Table:
| x | y |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
Each x appears once, so it’s a function.
Graph: use the vertical line test.
Equation: y = 2x + 1 is a function (one y for each x). x² + y² = 25 (a circle) is not.
The vertical line test
Draw (or imagine) vertical lines across a graph. If any vertical line crosses the graph more than once, it’s not a function.
| Graph | Function? |
|---|---|
| a straight non-vertical line | Yes |
| a parabola opening up | Yes |
| a circle | No |
| a vertical line | No |
| a sideways parabola (x = y²) | No |
Real-life functions
- The cost of gas depends on the litres bought (one cost per amount).
- Temperature at each time of day (one temperature per moment).
- Not a function: a person’s age → their friends (one age, many friends).
Worked examples
Example 1: Is {(2, 3), (4, 3), (6, 3)} a function? Yes; each input has one output.
Example 2: Is {(5, 1), (5, 2)} a function? No; 5 has two outputs.
Example 3: Is y² = x a function? For x = 4, y = 2 or −2 → no.
Common mistakes
- Thinking repeated outputs break the rule. Only repeated inputs with different outputs do.
- Using a horizontal line test instead of vertical.
- Assuming every equation is a function.
Try it yourself
- Is {(0, 0), (1, 1), (−1, 1)} a function?
- Is {(3, 4), (3, −4)} a function?
- Does a horizontal line pass the vertical line test?
- Is a student’s name → their birth date a function?
Answers: 1) yes 2) no 3) yes 4) yes (each person has one birth date)
Quick recap
- A relation pairs inputs and outputs.
- A function gives exactly one output for each input.
- Use the vertical line test on graphs.