To determine whether a set of points represents a function, we use the definition of a function. In mathematical terms, a function is a relation in which each input (or x-coordinate) is associated with exactly one output (or y-coordinate).
First Group of Points: {(0, 1), (0, 5), (2, 6), (3, 3)}
- Here, we notice that the point (0, 1) and the point (0, 5) share the same x-coordinate (which is 0), but they have different y-coordinates (1 and 5 respectively).
- This means that for the input x = 0, we get two different outputs, y = 1 and y = 5.
Since there is an x-value that corresponds to more than one y-value, this group of points does not represent a function.
Second Group of Points: {(1, 4), (2, 7), (3, 1), (5, 7)}
- For this set, each x-coordinate is unique: 1, 2, 3, and 5.
- Each x-value maps to one and only one y-value, regardless of whether some y-values are repeated.
Since every x-coordinate in this group has a unique y-coordinate, this set of points represents a function.
### Conclusion:
- The key observation that distinguishes a function from a non-function in these groups of points is the uniqueness of the x-values.
- If there is any repeated x-value in the set of points with different corresponding y-values, then it cannot be called a function.
- Conversely, if each x-value in the set corresponds to exactly one y-value, then it represents a function.
So, being a function means establishing a relationship where each input (x-value) has one and only one output (y-value).
