Determine which of the following relations is a function.
Answer (1)
A relation is a function if each x-value has only one corresponding y-value.
The first relation is a function because each x-value has a unique y-value.
The second and fourth relations are not functions because at least one x-value has multiple y-values.
The third relation is a function because each x-value has a unique y-value.
Explanation
Understanding Functions We are given four relations defined by tables of x and y values. A relation is a function if each x-value is associated with only one y-value. We need to determine which of the four given relations is a function.
Analyzing Relation 1 For the first relation:
| x | y |
| --- | --- |
| -10 | 10 |
| -5 | 5 |
| 0 | 0 |
| 5 | -5 |
| 10 | -10 |
Each x-value is associated with only one y-value. So, this is a function.
Analyzing Relation 2 For the second relation:
| x | y |
| --- | --- |
| -3 | 2 |
| -1 | 1 |
| 0 | 2 |
| 0 | 1 |
| 3 | 4 |
The x-value 0 is associated with two different y-values (2 and 1). So, this is not a function.
Analyzing Relation 3 For the third relation:
| x | y |
| --- | --- |
| -8 | -4 |
| -2 | -2 |
| 1 | 3 |
| 2 | 4 |
| 4 | 6 |
Each x-value is associated with only one y-value. So, this is a function.
Analyzing Relation 4 For the fourth relation:
| x | y |
| --- | --- |
| 0 | 2 |
| 1 | 4 |
| 2 | 2 |
| 2 | 6 |
| 5 | 7 |
The x-value 2 is associated with two different y-values (2 and 6). So, this is not a function.
Conclusion Therefore, the first and third relations are functions.
Examples
Functions are essential in modeling real-world relationships. For example, the relationship between the number of hours worked and the amount earned (assuming a fixed hourly wage) can be represented as a function. Similarly, the relationship between the radius of a circle and its area is a function. Understanding functions helps us predict and analyze various phenomena in science, engineering, and economics.
