The Image Of A Point Is Given By The Rule $r_{y=-x}(x, Y) \rightarrow(-4, 9$\]. What Are The Coordinates Of Its Pre-image?A. $(-9, 4$\]B. $(-4, -9$\]C. $(4, 9$\]D. $(9, -4$\]

Feb 27, 2025 by ADMIN 175 views

The Image of a Point: Understanding the Concept of Pre-Image and Image in Mathematics

In mathematics, particularly in geometry and coordinate geometry, the concept of image and pre-image plays a crucial role in understanding transformations and mappings. The image of a point is the point that results from applying a transformation to the original point, while the pre-image is the original point that is mapped to the image point. In this article, we will explore the concept of pre-image and image, and use a specific example to illustrate the process of finding the coordinates of the pre-image.

What is a Pre-Image?

A pre-image is the original point that is mapped to the image point under a given transformation. In other words, it is the point that is transformed into the image point. The pre-image is also known as the original point or the input point.

What is an Image?

An image is the point that results from applying a transformation to the original point. It is the point that is mapped to by the pre-image under the given transformation.

The Rule of Reflection

The rule of reflection is a specific type of transformation that involves reflecting a point across a line or a point. In this case, we are given the rule $r_{y=-x}(x, y) \rightarrow(-4, 9)$ , which means that the point $(x, y)$ is reflected across the line $y = -x$ to obtain the image point $(-4, 9)$ .

Finding the Pre-Image

To find the pre-image of the point $(-4, 9)$ , we need to apply the inverse transformation of the rule of reflection. Since the rule of reflection involves reflecting a point across the line $y = -x$ , the inverse transformation will involve reflecting the image point across the same line.

Step 1: Reflecting the Image Point

To reflect the image point $(-4, 9)$ across the line $y = -x$ , we need to find the point that is symmetric to $(-4, 9)$ with respect to the line $y = -x$ . This can be done by swapping the $x$ and $y$ coordinates of the image point and changing the sign of one of the coordinates.

Step 2: Finding the Pre-Image

After reflecting the image point $(-4, 9)$ across the line $y = -x$ , we obtain the pre-image point $(9, -4)$ .

In conclusion, the pre-image of the point $(-4, 9)$ under the rule of reflection $r_{y=-x}(x, y) \rightarrow(-4, 9)$ is the point $(9, -4)$ . This can be verified by applying the inverse transformation of the rule of reflection to the image point $(-4, 9)$ .

The correct answer is:

D. $(9, -4)$

The concept of pre-image and image is a fundamental idea in mathematics, particularly in geometry and coordinate geometry. Understanding the relationship between pre-image and image is crucial in applying transformations and mappings to solve problems in mathematics and other fields.

Find the pre-image of the point $(2, 3)$ under the rule of reflection $r_{y=x}(x, y) \rightarrow(3, 2)$ .
Find the image of the point $(4, 5)$ under the rule of reflection $r_{y=-x}(x, y) \rightarrow(-5, 4)$ .
Find the pre-image of the point $(-2, 1)$ under the rule of reflection $r_{y=x}(x, y) \rightarrow(1, -2)$ .

The pre-image of the point $(2, 3)$ under the rule of reflection $r_{y=x}(x, y) \rightarrow(3, 2)$ is the point $(3, 2)$ .
The image of the point $(4, 5)$ under the rule of reflection $r_{y=-x}(x, y) \rightarrow(-5, 4)$ is the point $(-5, 4)$ .
The pre-image of the point $(-2, 1)$ under the rule of reflection $r_{y=x}(x, y) \rightarrow(1, -2)$ is the point $(1, -2)$ .

In conclusion, the concept of pre-image and image is a fundamental idea in mathematics, particularly in geometry and coordinate geometry. Understanding the relationship between pre-image and image is crucial in applying transformations and mappings to solve problems in mathematics and other fields.
Q&A: Understanding Pre-Image and Image in Mathematics

In our previous article, we explored the concept of pre-image and image in mathematics, particularly in geometry and coordinate geometry. We discussed the rule of reflection and how to find the pre-image of a point under a given transformation. In this article, we will answer some frequently asked questions about pre-image and image to help you better understand this concept.

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

A: The pre-image is the original point that is mapped to the image point under a given transformation. The image is the point that results from applying a transformation to the original point.

Q: How do I find the pre-image of a point under a given transformation?

A: To find the pre-image of a point under a given transformation, you need to apply the inverse transformation of the rule of reflection. This involves reflecting the image point across the same line or point that was used to obtain the image point.

Q: What is the rule of reflection?

A: The rule of reflection is a specific type of transformation that involves reflecting a point across a line or a point. In this case, we are given the rule $r_{y=-x}(x, y) \rightarrow(-4, 9)$ , which means that the point $(x, y)$ is reflected across the line $y = -x$ to obtain the image point $(-4, 9)$ .

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

A: To apply the rule of reflection, you need to swap the $x$ and $y$ coordinates of the image point and change the sign of one of the coordinates. This will give you the pre-image point.

Q: What is the inverse transformation of the rule of reflection?

A: The inverse transformation of the rule of reflection is the transformation that is applied to the image point to obtain the pre-image point. In this case, the inverse transformation is the reflection of the image point across the line $y = -x$ .

Q: How do I find the pre-image of a point under a given transformation if the transformation is not a reflection?

A: If the transformation is not a reflection, you need to use the inverse function of the transformation to find the pre-image of the point. This involves solving the equation for the original point $(x, y)$ in terms of the image point $(x', y')$ .

Q: What are some common transformations that involve pre-image and image?

A: Some common transformations that involve pre-image and image include:

Reflections across lines or points
Rotations about points or axes
Translations of points or objects
Scaling of points or objects

Q: How do I apply transformations to solve problems in mathematics and other fields?

A: To apply transformations to solve problems in mathematics and other fields, you need to understand the concept of pre-image and image and how to apply the inverse transformation to find the pre-image of a point. You also need to be able to use the transformation to solve equations and inequalities.

In conclusion, the concept of pre-image and image is a fundamental idea in mathematics, particularly in geometry and coordinate geometry. Understanding the relationship between pre-image and image is crucial in applying transformations and mappings to solve problems in mathematics and other fields. By answering these frequently asked questions, we hope to have provided you with a better understanding of this concept and how to apply it to solve problems.