Question

select all of the ordered pairs that could be added to the graph so that it still shows a relation that is a function. (2, 6) (3, 8) (7, 10) (9, 4)

Explanation:

Step1: Recall Function Definition

A function is a relation where each input (x - value) has exactly one output (y - value). So, we need to check if the x - value of each ordered pair is already used in the existing graph.
Existing x - values from the graph: Looking at the points, the x - values are 1, 4, 9 (from the points (1,1), (4,3), (9,2)).

Step2: Check Each Ordered Pair

- For (2, 6): The x - value is 2, which is not in {1, 4, 9}, so it can be added.
- For (3, 8): The x - value is 3, which is not in {1, 4, 9}, so it can be added.
- For (7, 10): The x - value is 7, which is not in {1, 4, 9}, so it can be added.
- For (9, 4): The x - value is 9, which is already in {1, 4, 9} (since there is a point at x = 9 with y = 2). Wait, no, wait. Wait, the existing point at x = 9 has y = 2. If we add (9, 4), then x = 9 would have two y - values (2 and 4), which violates the function definition. Wait, maybe I misread the graph. Wait, let's re - examine. Wait, the original graph: the points are (1,1), (4,3), (9,2). So x - values are 1, 4, 9. So for (9,4), x = 9 is already used. But in the checkboxes, (9,4) is checked. Wait, maybe I made a mistake. Wait, no, wait the problem says "select all of the ordered pairs that could be added so that it still shows a relation that is a function". So a function requires that each x has at most one y. So if an x is not used, we can add any y. If an x is used, we can't add another pair with the same x. So let's check each:

- (2,6): x = 2 is not used (original x's: 1,4,9), so okay.
- (3,8): x = 3 is not used, okay.
- (7,10): x = 7 is not used, okay.
- (9,4): x = 9 is used (original point (9,2)). So adding (9,4) would mean x = 9 has two y - values (2 and 4), which is not a function. Wait, but the checkbox for (9,4) is checked. Wait, maybe I misread the original graph. Wait, let's look again. The x - axis labels: 1,2,3,4,5,6,7,8,9,10. The points: at x = 1, y = 1; x = 4, y = 3; x = 9, y = 2. So x = 9 is already taken. So (9,4) should not be added. But the problem's checkboxes show (9,4) as checked. Wait, maybe I made a mistake. Wait, no, the key is: a function is defined as each x has unique y. So if x is new, any y is fine. If x is old, y must be the same. So:

Wait, maybe the original graph's x - values: let's list the coordinates properly. The first point: x = 1, y = 1 (since it's at x = 1, y = 1). Second: x = 4, y = 3 (x = 4, y = 3). Third: x = 9, y = 2 (x = 9, y = 2). So x - values are 1,4,9. So:

- (2,6): x = 2 (not in {1,4,9}) → valid.
- (3,8): x = 3 (not in {1,4,9}) → valid.
- (7,10): x = 7 (not in {1,4,9}) → valid.
- (9,4): x = 9 (in {1,4,9}) → invalid, because x = 9 already has y = 2. So adding (9,4) would make x = 9 map to two y - values, which is not a function. But the problem's checkboxes have (9,4) checked. Maybe there's a mistake in my analysis. Wait, maybe the original graph's x - values are different. Wait, maybe the x - coordinates: let's see the grid. The first point is at x = 1 (column 1), second at x = 4 (column 4), third at x = 9 (column 9). So x = 9 is used. So (9,4) is invalid. But the problem's checkboxes include (9,4). Wait, maybe the question is wrong, or maybe I misread. Wait, no, the task is to check which can be added. So the correct ones are (2,6), (3,8), (7,10), because their x - values are not already used. (9,4) has x = 9 which is already used, so it should not be added. But according to the checkboxes in the image, maybe the original graph's x - values are different. Wait, maybe I made a mistake in identifying the original x - values. Let's re - check the graph:

The graph has points at x = 1 (y = 1), x = 4 (y = 3), x = 9 (y = 2). So x - values: 1,4,9. So:

- (2,6): x = 2 (new) → okay.
- (3,8): x = 3 (new) → okay.
- (7,10): x = 7 (new) → okay.
- (9,4): x = 9 (old) → not okay. But the problem's checkboxes show (9,4) as checked. Maybe the question has a mistake, but according to the function definition, (2,6), (3,8), (7,10) are valid, (9,4) is not. But maybe I misread the original points. Wait, maybe the third point is at x = 10? No, the x - axis goes to 10, and the third point is at x = 9 (since it's at the column labeled 9). So, the correct ordered pairs that can be added are (2,6), (3,8), (7,10) because their x - values are not already present, so each x will have only one y - value. (9,4) has an x - value that's already present, so it can't be added without violating the function definition. But the image shows (9,4) as checked. Maybe the original graph's x - value for the third point is different. Wait, maybe the third point is at x = 10? No, the x - label at the third point is 9. So, perhaps the intended answer is (2,6), (3,8), (7,10), and (9,4) is a mistake. But according to the function definition, the x - values of the new ordered pairs must not be in {1,4,9}. So (2,6) (x = 2), (3,8) (x = 3), (7,10) (x = 7) are valid, (9,4) (x = 9) is invalid.

Answer:

The ordered pairs that can be added to form a function are (2, 6), (3, 8), (7, 10). (Note: (9, 4) should not be added as it violates the function definition since x = 9 already has a y - value.) But if we follow the checkboxes in the image (assuming a possible error in my analysis), the checked ones are (2, 6), (3, 8), (7, 10), (9, 4) – but (9, 4) is incorrect. However, based on the function definition, the correct ones are (2, 6), (3, 8), (7, 10).