Finding A Pre-Image Using A RuleThe Image Of A Point Is Given By The Rule R Y = − X ( X , Y ) → ( − 4 , 9 R_y = -x(x, Y) \rightarrow (-4, 9 R Y ​ = − X ( X , Y ) → ( − 4 , 9 ]. What Are The Coordinates Of Its Pre-image?A. ( − 9 , 4 (-9, 4 ( − 9 , 4 ]B. ( − 4 , − 9 (-4, -9 ( − 4 , − 9 ]C. ( 4 , 9 (4, 9 ( 4 , 9 ]D. ( 9 , − 4 (9, -4 ( 9 , − 4 ]

Introduction

In mathematics, a pre-image is a point in the original domain that is mapped to a specific point in the range by a given function or rule. In this article, we will explore how to find the pre-image of a point using a rule, with a focus on the given image of a point defined by the rule $r_y = -x(x, y) \rightarrow (-4, 9)$. We will delve into the mathematical concepts and provide step-by-step solutions to find the coordinates of the pre-image.

Understanding the Rule

The given rule is $r_y = -x(x, y) \rightarrow (-4, 9)$. This rule states that for every point $(x, y)$ in the original domain, the corresponding image point is $(-4, 9)$. To find the pre-image, we need to reverse this process and determine the original point $(x, y)$ that maps to the given image point $(-4, 9)$.

Step 1: Analyze the Rule

Let's analyze the rule $r_y = -x(x, y) \rightarrow (-4, 9)$. We can see that the rule involves a simple transformation of the x-coordinate, where $x$ is multiplied by $-1$ to obtain the image x-coordinate. This means that the pre-image x-coordinate will be the negative of the image x-coordinate.

Step 2: Find the Pre-Image X-Coordinate

Using the analysis from Step 1, we can find the pre-image x-coordinate by taking the negative of the image x-coordinate. In this case, the image x-coordinate is $-4$, so the pre-image x-coordinate is:

$$x = -(-4) = 4$$

Step 3: Find the Pre-Image Y-Coordinate

Now that we have the pre-image x-coordinate, we need to find the pre-image y-coordinate. Since the rule involves a simple transformation of the x-coordinate, the y-coordinate remains unchanged. Therefore, the pre-image y-coordinate is the same as the image y-coordinate, which is $9$.

Conclusion

In conclusion, the coordinates of the pre-image are $(4, 9)$. This means that the original point $(x, y)$ that maps to the given image point $(-4, 9)$ is $(4, 9)$.

Answer

The correct answer is:

• C. $(4, 9)$

Discussion

This problem involves a simple rule that maps points from the original domain to the range. By analyzing the rule and reversing the process, we can find the pre-image of a given point. This type of problem is essential in mathematics, as it helps us understand the concept of pre-images and how to work with functions and rules.

Real-World Applications

The concept of pre-images has numerous real-world applications in fields such as computer graphics, image processing, and data analysis. For example, in computer graphics, pre-images are used to create 3D models and animations by mapping points from the original domain to the range. In image processing, pre-images are used to restore images that have been distorted or corrupted.

Conclusion

Introduction

In our previous article, we explored how to find the pre-image of a point using a rule. We analyzed the given rule, found the pre-image x-coordinate, and determined the pre-image y-coordinate. In this article, we will provide a Q&A guide to help you better understand the concept of pre-images and how to work with functions and rules.

Q: What is a pre-image?

A: A pre-image is a point in the original domain that is mapped to a specific point in the range by a given function or rule.

Q: How do I find the pre-image of a point using a rule?

A: To find the pre-image of a point using a rule, you need to analyze the rule, find the pre-image x-coordinate, and determine the pre-image y-coordinate.

Q: What is the difference between a pre-image and an image?

A: A pre-image is a point in the original domain that is mapped to a specific point in the range, while an image is the point in the range that is mapped to by a point in the original domain.

Q: Can I use the same rule to find the pre-image of multiple points?

A: Yes, you can use the same rule to find the pre-image of multiple points. However, you need to make sure that the rule is consistent and that the pre-image points are correctly determined.

Q: How do I know if a rule is consistent?

A: A rule is consistent if it maps points from the original domain to the range in a predictable and consistent manner. You can check the consistency of a rule by applying it to multiple points and verifying that the pre-image points are correctly determined.

Q: What are some common types of rules that I can use to find pre-images?

A: Some common types of rules that you can use to find pre-images include:

• Linear rules: These rules involve a simple transformation of the x-coordinate, such as multiplication or addition.
• Non-linear rules: These rules involve a more complex transformation of the x-coordinate, such as squaring or taking the square root.
• Piecewise rules: These rules involve a combination of different transformations, such as linear and non-linear transformations.

Q: How do I apply a rule to find the pre-image of a point?

A: To apply a rule to find the pre-image of a point, you need to follow these steps:

1. Analyze the rule and determine the type of transformation that is involved.
2. Find the pre-image x-coordinate by applying the transformation to the image x-coordinate.
3. Determine the pre-image y-coordinate by applying the transformation to the image y-coordinate.
4. Verify that the pre-image point is correctly determined by applying the rule to the pre-image point.

Q: What are some common mistakes to avoid when finding pre-images?

A: Some common mistakes to avoid when finding pre-images include:

• Failing to analyze the rule and determine the type of transformation that is involved.
• Making errors when applying the transformation to find the pre-image x-coordinate or y-coordinate.
• Failing to verify that the pre-image point is correctly determined.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concept of pre-images and how to work with functions and rules. We covered common types of rules, how to apply a rule to find the pre-image of a point, and common mistakes to avoid. By following these guidelines, you can improve your understanding of pre-images and become more confident in your ability to work with functions and rules.