Determine which of the following relations is a function.
| x | y |
| --- | --- |
| -10 | 10 |
| -5 | 5 |
| 0 | 0 |
| 5 | -5 |
| 10 | -10 |
| x | y |
| ----- | --- |
| -3 | 2 |
| -1 | 1 |
| 0 | 2 |
| 0 | 1 |
| 3 | 4 |
| x | y |
| ----- | --- |
| -8 | -4 |
| 5 h-2 | -2 |
| 1 | 3 |
| 2 | 4 |
| 4 | 6 |
| x | y |
| --- | --- |
| 0 | 2 |
| 1 | 4 |
| 2 | 2 |
| 2 | 6 |
| 5 | 7 |
Answer (1)
A relation is a function if each x -value has only one y -value.
Relation 1: Each x has one y , so it is a function.
Relation 2: x = 0 has two y values, so it is not a function.
Relation 3: Each x has one y , so it is a function.
Relation 4: x = 2 has two y values, so it is not a function.
Explanation
Understanding Functions A relation is a function if each x -value is associated with only one y -value. We will examine each relation to determine if it is a function.
Analyzing Relation 1 In the first relation, we have the points ( − 10 , 10 ) , ( − 5 , 5 ) , ( 0 , 0 ) , ( 5 , − 5 ) , and ( 10 , − 10 ) . Each x -value is associated with exactly one y -value. Therefore, this relation is a function.
Analyzing Relation 2 In the second relation, we have the points ( − 3 , 2 ) , ( − 1 , 1 ) , ( 0 , 2 ) , ( 0 , 1 ) , and ( 3 , 4 ) . The x -value 0 is associated with two y -values, 2 and 1 . Therefore, this relation is not a function.
Analyzing Relation 3 In the third relation, we have the points ( − 8 , − 4 ) , ( 5 h − 2 , − 2 ) , ( 1 , 3 ) , ( 2 , 4 ) , and ( 4 , 6 ) . Each x -value is associated with exactly one y -value. Note that 5 h − 2 represents a single x -value. Therefore, this relation is a function.
Analyzing Relation 4 In the fourth relation, we have the points ( 0 , 2 ) , ( 1 , 4 ) , ( 2 , 2 ) , ( 2 , 6 ) , and ( 5 , 7 ) . The x -value 2 is associated with two y -values, 2 and 6 . Therefore, this relation is not a function.
Conclusion The first and third relations are functions, while the second and fourth relations are not functions.
Examples
Functions are essential in modeling real-world relationships where each input has a unique output. For example, consider a vending machine where each button (input) corresponds to a specific snack (output). If pressing the same button sometimes gives you different snacks, the vending machine is not functioning properly, similar to how a relation that is not a function behaves. Understanding functions helps us ensure predictability and reliability in various systems.
