Will The Relation $y=6-\sqrt{2x+74}$ Pass The Vertical Line Test?

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Introduction

The vertical line test is a method used to determine whether a relation is a function or not. In this test, we draw a vertical line on the graph of the relation and check if it intersects the graph at more than one point. If the vertical line intersects the graph at more than one point, then the relation is not a function and does not pass the vertical line test. In this article, we will explore whether the relation $y=6-\sqrt{2x+74}$ passes the vertical line test.

Understanding the Vertical Line Test

The vertical line test is a graphical method used to determine whether a relation is a function or not. A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). In other words, a function is a relation where each input corresponds to exactly one output. The vertical line test is used to check if a relation satisfies this condition.

To perform the vertical line test, we draw a vertical line on the graph of the relation. If the vertical line intersects the graph at more than one point, then the relation is not a function and does not pass the vertical line test. On the other hand, if the vertical line intersects the graph at exactly one point, then the relation is a function and passes the vertical line test.

Analyzing the Relation

The given relation is $y=6-\sqrt{2x+74}$. To determine whether this relation passes the vertical line test, we need to analyze its graph. The graph of this relation is a curve that opens downwards. To understand why this curve opens downwards, let's consider the square root term in the relation.

The square root term is $\sqrt{2x+74}$. This term is always non-negative, since the square root of a number is always non-negative. As x increases, the value of $\sqrt{2x+74}$ also increases. Therefore, the value of $6-\sqrt{2x+74}$ decreases as x increases.

Graphical Analysis

To analyze the graph of the relation, let's consider the following points:

  • When x = -37, the value of $\sqrt{2x+74}$ is 0. Therefore, the value of $6-\sqrt{2x+74}$ is 6.
  • When x = -36, the value of $\sqrt{2x+74}$ is 2. Therefore, the value of $6-\sqrt{2x+74}$ is 4.
  • When x = -35, the value of $\sqrt{2x+74}$ is 4. Therefore, the value of $6-\sqrt{2x+74}$ is 2.
  • When x = -34, the value of $\sqrt{2x+74}$ is 6. Therefore, the value of $6-\sqrt{2x+74}$ is 0.

From these points, we can see that the graph of the relation is a curve that opens downwards. The curve intersects the x-axis at x = -37, x = -34, and other points. Therefore, the vertical line test is not satisfied, and the relation does not pass the vertical line test.

Conclusion

In conclusion, the relation $y=6-\sqrt{2x+74}$ does not pass the vertical line test. This is because the graph of the relation is a curve that opens downwards and intersects the x-axis at multiple points. Therefore, the vertical line test is not satisfied, and the relation is not a function.

Final Thoughts

The vertical line test is a useful tool for determining whether a relation is a function or not. In this article, we analyzed the relation $y=6-\sqrt{2x+74}$ and determined that it does not pass the vertical line test. This is because the graph of the relation is a curve that opens downwards and intersects the x-axis at multiple points. Therefore, the vertical line test is not satisfied, and the relation is not a function.

References

  • [1] "Functions and Relations" by Khan Academy
  • [2] "Vertical Line Test" by Math Open Reference
  • [3] "Graphing Functions" by Purplemath

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Introduction

In our previous article, we analyzed the relation $y=6-\sqrt{2x+74}$ and determined that it does not pass the vertical line test. In this article, we will answer some frequently asked questions related to the vertical line test and the relation $y=6-\sqrt{2x+74}$.

Q&A

Q: What is the vertical line test?

A: The vertical line test is a method used to determine whether a relation is a function or not. In this test, we draw a vertical line on the graph of the relation and check if it intersects the graph at more than one point. If the vertical line intersects the graph at more than one point, then the relation is not a function and does not pass the vertical line test.

Q: Why is the vertical line test important?

A: The vertical line test is important because it helps us determine whether a relation is a function or not. A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). In other words, a function is a relation where each input corresponds to exactly one output. The vertical line test is used to check if a relation satisfies this condition.

Q: How do I perform the vertical line test?

A: To perform the vertical line test, follow these steps:

  1. Draw a vertical line on the graph of the relation.
  2. Check if the vertical line intersects the graph at more than one point.
  3. If the vertical line intersects the graph at more than one point, then the relation is not a function and does not pass the vertical line test.

Q: What is the graph of the relation $y=6-\sqrt{2x+74}$ like?

A: The graph of the relation $y=6-\sqrt{2x+74}$ is a curve that opens downwards. The curve intersects the x-axis at multiple points.

Q: Why does the graph of the relation $y=6-\sqrt{2x+74}$ open downwards?

A: The graph of the relation $y=6-\sqrt{2x+74}$ opens downwards because the square root term $\sqrt{2x+74}$ is always non-negative. As x increases, the value of $\sqrt{2x+74}$ also increases. Therefore, the value of $6-\sqrt{2x+74}$ decreases as x increases.

Q: Does the relation $y=6-\sqrt{2x+74}$ pass the vertical line test?

A: No, the relation $y=6-\sqrt{2x+74}$ does not pass the vertical line test. This is because the graph of the relation is a curve that opens downwards and intersects the x-axis at multiple points.

Q: What are some common mistakes to avoid when performing the vertical line test?

A: Some common mistakes to avoid when performing the vertical line test include:

  • Drawing a horizontal line instead of a vertical line.
  • Not checking if the vertical line intersects the graph at more than one point.
  • Not considering the domain and range of the relation.

Q: How can I use the vertical line test in real-life situations?

A: The vertical line test can be used in real-life situations such as:

  • Determining whether a relation is a function or not.
  • Analyzing the behavior of a relation.
  • Identifying the domain and range of a relation.

Conclusion

In conclusion, the vertical line test is a useful tool for determining whether a relation is a function or not. In this article, we answered some frequently asked questions related to the vertical line test and the relation $y=6-\sqrt{2x+74}$. We hope that this article has provided you with a better understanding of the vertical line test and how to use it in real-life situations.

Final Thoughts

The vertical line test is a simple yet powerful tool for analyzing relations. By using the vertical line test, we can determine whether a relation is a function or not and gain a deeper understanding of its behavior. We hope that this article has inspired you to explore the world of functions and relations and to use the vertical line test in your own mathematical adventures.

References

  • [1] "Functions and Relations" by Khan Academy
  • [2] "Vertical Line Test" by Math Open Reference
  • [3] "Graphing Functions" by Purplemath

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