To ensure that a relation is a function, every x-value (also known as input or domain) must be matched with exactly one y-value (output or range). In other words, there cannot be multiple y-values associated with the same x-value within the context of the function.
Now, let's look at the set of ordered pairs given in the relation:
```
((3, 0), (1, 4), (5, 9), (2, 8), (x, y))
```
The x-values in this set of ordered pairs are 3, 1, 5, and 2. For the relation to remain a function with the addition of the new ordered pair `(x, y)`, the x-value of this new pair must not duplicate any of the x-values that are already present.
Let's go through the given options for `(x, y)` to determine which one can be added without violating the definition of a function:
Option (2,7):
Since the x-value 2 is already in the set (from the ordered pair (2, 8)), including this option would result in two different y-values (7 and 8) for the same x-value (2). Thus, this option would not satisfy the condition for the relation to be a function.
Option (1,5):
The x-value 1 is also already in the set (from the ordered pair (1, 4)), including this option would give us another pair with the x-value 1 but a different y-value. Thus, having (1,4) and (1,5) would violate the requirement for a function, so this option is also not correct.
Option (3,3):
The x-value 3 is already in the set as well (from the ordered pair (3, 0)), and adding this option would mean having the same x-value with a different y-value, which does not satisfy the condition for the relation to be a function.
Option (0,0):
This option has an x-value of 0, which is not present in the existing set of x-values. So, adding the ordered pair (0, 0) would not result in duplicate x-values and therefore would not violate the function's condition. This is the correct ordered pair that could be added to ensure that the relation remains a function.
In conclusion, the ordered pair that can be substituted for `(x, y)` to ensure that the relation is a function is (0,0).
