Which Ordered Pair Could Be Removed To Make This Relation A Function?A. { (-5,0)$}$ B. { (-1,-3)$}$ C. { (4,-2)$}$ D. { (6,-1)$}$

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Understanding Functions and Relations

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function is a special type of relation where each input corresponds to exactly one output. In other words, for every input, there is only one possible output. On the other hand, a relation is a set of ordered pairs that do not necessarily have a one-to-one correspondence between the inputs and outputs.

The Importance of Removing Ordered Pairs

When we have a relation, we may need to remove some ordered pairs to make it a function. This is because a relation can have multiple outputs for the same input, which is not allowed in a function. By removing the ordered pairs that cause this issue, we can create a function from the relation.

Analyzing the Given Relation

The given relation is not explicitly stated, but we are given four ordered pairs: A. (−5,0){(-5,0)}, B. (−1,−3){(-1,-3)}, C. (4,−2){(4,-2)}, and D. (6,−1){(6,-1)}. We need to determine which of these ordered pairs could be removed to make the relation a function.

Checking for Duplicate Inputs

To make the relation a function, we need to ensure that each input corresponds to exactly one output. Let's check if any of the given ordered pairs have the same input.

  • A. (−5,0){(-5,0)} has an input of -5.
  • B. (−1,−3){(-1,-3)} has an input of -1.
  • C. (4,−2){(4,-2)} has an input of 4.
  • D. (6,−1){(6,-1)} has an input of 6.

We can see that none of the ordered pairs have the same input. Therefore, we cannot remove any of the ordered pairs based on duplicate inputs.

Checking for Multiple Outputs

Now, let's check if any of the ordered pairs have the same input and a different output. If we find such an ordered pair, we can remove it to make the relation a function.

  • A. (−5,0){(-5,0)} has an input of -5 and an output of 0.
  • B. (−1,−3){(-1,-3)} has an input of -1 and an output of -3.
  • C. (4,−2){(4,-2)} has an input of 4 and an output of -2.
  • D. (6,−1){(6,-1)} has an input of 6 and an output of -1.

We can see that none of the ordered pairs have the same input and a different output. Therefore, we cannot remove any of the ordered pairs based on multiple outputs.

Conclusion

Based on our analysis, we cannot remove any of the ordered pairs A. (−5,0){(-5,0)}, B. (−1,−3){(-1,-3)}, C. (4,−2){(4,-2)}, or D. (6,−1){(6,-1)} to make the relation a function. However, we can try removing each of these ordered pairs one by one and see if the resulting relation is a function.

Removing Ordered Pair A

Let's remove ordered pair A. (−5,0){(-5,0)} from the relation. The resulting relation is:

(−1,−3){(-1,-3)} (4,−2){(4,-2)} (6,−1){(6,-1)}

We can see that the resulting relation is a function because each input corresponds to exactly one output.

Removing Ordered Pair B

Let's remove ordered pair B. (−1,−3){(-1,-3)} from the relation. The resulting relation is:

(−5,0){(-5,0)} (4,−2){(4,-2)} (6,−1){(6,-1)}

We can see that the resulting relation is a function because each input corresponds to exactly one output.

Removing Ordered Pair C

Let's remove ordered pair C. (4,−2){(4,-2)} from the relation. The resulting relation is:

(−5,0){(-5,0)} (−1,−3){(-1,-3)} (6,−1){(6,-1)}

We can see that the resulting relation is a function because each input corresponds to exactly one output.

Removing Ordered Pair D

Let's remove ordered pair D. (6,−1){(6,-1)} from the relation. The resulting relation is:

(−5,0){(-5,0)} (−1,−3){(-1,-3)} (4,−2){(4,-2)}

We can see that the resulting relation is a function because each input corresponds to exactly one output.

Conclusion

Based on our analysis, we can remove any of the ordered pairs A. (−5,0){(-5,0)}, B. (−1,−3){(-1,-3)}, C. (4,−2){(4,-2)}, or D. (6,−1){(6,-1)} to make the relation a function. However, we can only remove one ordered pair at a time. Therefore, the correct answer is the ordered pair that, when removed, results in a function.

The Correct Answer

The correct answer is B. (−1,−3){(-1,-3)}. When we remove this ordered pair, the resulting relation is a function.

Final Answer

Q: What is a function in mathematics?

A: A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function is a special type of relation where each input corresponds to exactly one output.

Q: What is a relation in mathematics?

A: A relation is a set of ordered pairs that do not necessarily have a one-to-one correspondence between the inputs and outputs.

Q: Why do we need to remove ordered pairs to make a relation a function?

A: We need to remove ordered pairs to make a relation a function because a relation can have multiple outputs for the same input, which is not allowed in a function.

Q: How do we determine which ordered pair to remove to make a relation a function?

A: We can determine which ordered pair to remove by checking if any of the ordered pairs have the same input or the same input and a different output. If we find such an ordered pair, we can remove it to make the relation a function.

Q: What happens if we remove an ordered pair that has the same input and a different output?

A: If we remove an ordered pair that has the same input and a different output, the resulting relation will be a function because each input will correspond to exactly one output.

Q: What happens if we remove an ordered pair that has the same input but a different output is not present?

A: If we remove an ordered pair that has the same input but a different output is not present, the resulting relation will still not be a function because the input will still correspond to multiple outputs.

Q: Can we remove any ordered pair to make a relation a function?

A: No, we cannot remove any ordered pair to make a relation a function. We need to remove an ordered pair that has the same input and a different output.

Q: How do we know which ordered pair to remove to make a relation a function?

A: We can know which ordered pair to remove by analyzing the given relation and checking if any of the ordered pairs have the same input or the same input and a different output.

Q: What is the correct answer to the problem?

A: The correct answer is B. (−1,−3){(-1,-3)}. When we remove this ordered pair, the resulting relation is a function.

Q: Why is B. (−1,−3){(-1,-3)} the correct answer?

A: B. (−1,−3){(-1,-3)} is the correct answer because when we remove this ordered pair, the resulting relation is a function. The other ordered pairs do not have the same input and a different output, so removing them would not result in a function.

Q: What is the final answer to the problem?

A: The final answer is B.