Which Ordered Pair Can Be Removed So That The Resulting Graph Represents A Function?A. ( − 2 , 2 (-2,2 ( − 2 , 2 ] B. ( 1 , 3 (1,3 ( 1 , 3 ] C. ( 5 , − 4 (5,-4 ( 5 , − 4 ] D. ( − 4 , − 4 (-4,-4 ( − 4 , − 4 ]

by ADMIN 211 views

Understanding Functions and Graphs

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It is a way of describing a relationship between variables, where each input corresponds to exactly one output. A graph of a function is a visual representation of this relationship, where the x-axis represents the input and the y-axis represents the output.

Vertical Line Test

One way to determine if a graph represents a function is to use the vertical line test. This test states that if a vertical line intersects the graph at more than one point, then the graph does not represent a function. This is because a function must have a unique output for each input, and a vertical line represents a constant input.

Analyzing the Given Options

We are given four ordered pairs: A. (−2,2)(-2,2), B. (1,3)(1,3), C. (5,−4)(5,-4), and D. (−4,−4)(-4,-4). We need to determine which of these pairs can be removed so that the resulting graph represents a function.

Option A: (−2,2)(-2,2)

If we remove the point (−2,2)(-2,2) from the graph, we are left with three points: (1,3)(1,3), (5,−4)(5,-4), and (−4,−4)(-4,-4). To determine if this graph represents a function, we can use the vertical line test. If we draw a vertical line at x=−2x=-2, it will not intersect the graph at more than one point. Therefore, removing the point (−2,2)(-2,2) does not affect the graph's ability to represent a function.

Option B: (1,3)(1,3)

If we remove the point (1,3)(1,3) from the graph, we are left with three points: (−2,2)(-2,2), (5,−4)(5,-4), and (−4,−4)(-4,-4). To determine if this graph represents a function, we can use the vertical line test. If we draw a vertical line at x=1x=1, it will intersect the graph at more than one point. Therefore, removing the point (1,3)(1,3) does not guarantee that the resulting graph will represent a function.

Option C: (5,−4)(5,-4)

If we remove the point (5,−4)(5,-4) from the graph, we are left with three points: (−2,2)(-2,2), (1,3)(1,3), and (−4,−4)(-4,-4). To determine if this graph represents a function, we can use the vertical line test. If we draw a vertical line at x=5x=5, it will not intersect the graph at more than one point. Therefore, removing the point (5,−4)(5,-4) does not affect the graph's ability to represent a function.

Option D: (−4,−4)(-4,-4)

If we remove the point (−4,−4)(-4,-4) from the graph, we are left with three points: (−2,2)(-2,2), (1,3)(1,3), and (5,−4)(5,-4). To determine if this graph represents a function, we can use the vertical line test. If we draw a vertical line at x=−4x=-4, it will intersect the graph at more than one point. Therefore, removing the point (−4,−4)(-4,-4) does not guarantee that the resulting graph will represent a function.

Conclusion

Based on the analysis above, we can see that removing the point (−4,−4)(-4,-4) from the graph will result in a graph that represents a function. This is because the vertical line test will not intersect the graph at more than one point after removing this point.

Final Answer

The final answer is D. (−4,−4)(-4,-4).

Understanding Functions and Graphs

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It is a way of describing a relationship between variables, where each input corresponds to exactly one output. A graph of a function is a visual representation of this relationship, where the x-axis represents the input and the y-axis represents the output.

Vertical Line Test

One way to determine if a graph represents a function is to use the vertical line test. This test states that if a vertical line intersects the graph at more than one point, then the graph does not represent a function. This is because a function must have a unique output for each input, and a vertical line represents a constant input.

Q&A

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). It is a way of describing a relationship between variables, where each input corresponds to exactly one output.

Q: How can we determine if a graph represents a function?

A: We can use the vertical line test. If a vertical line intersects the graph at more than one point, then the graph does not represent a function.

Q: What is the vertical line test?

A: The vertical line test is a method used to determine if a graph represents a function. It states that if a vertical line intersects the graph at more than one point, then the graph does not represent a function.

Q: Why is the vertical line test important?

A: The vertical line test is important because it helps us determine if a graph represents a function. A function must have a unique output for each input, and the vertical line test helps us check for this uniqueness.

Q: Can a graph represent a function if it has multiple points with the same x-coordinate?

A: No, a graph cannot represent a function if it has multiple points with the same x-coordinate. This is because a function must have a unique output for each input, and multiple points with the same x-coordinate would indicate multiple outputs for the same input.

Q: How can we remove an ordered pair from a graph so that the resulting graph represents a function?

A: We can remove an ordered pair from a graph so that the resulting graph represents a function by identifying the point that causes the graph to fail the vertical line test. This point is the one that needs to be removed.

Q: What is the final answer to the original question?

A: The final answer to the original question is D. (−4,−4)(-4,-4). This is because removing the point (−4,−4)(-4,-4) from the graph will result in a graph that represents a function.

Conclusion

In conclusion, the vertical line test is an important tool for determining if a graph represents a function. By using this test, we can identify the points on a graph that cause it to fail the test and remove them to create a graph that represents a function. The final answer to the original question is D. (−4,−4)(-4,-4).

Final Answer

The final answer is D. (−4,−4)(-4,-4).