Applying the Inverse Transformation

Oct 13, 2025 by windailys.com 100 views

Understanding the Concept of Image and Pre-Image

In mathematics, particularly in geometry and coordinate geometry, the concept of image and pre-image plays a crucial role in understanding transformations. A transformation is a function that takes an input, called the pre-image, and produces an output, called the image. In this article, we will focus on the concept of image and pre-image under a given transformation rule.

The Transformation Rule

The transformation rule given is $r_{y=-x}(x, y) = (-4, 9)$ . This rule states that the image of a point $(x, y)$ under the transformation $r_{y=-x}$ is the point $(-4, 9)$ . The transformation $r_{y=-x}$ is a reflection across the line $y = -x$ .

Finding the Pre-Image

To find the pre-image of the point $(-4, 9)$ , we need to apply the inverse transformation of $r_{y=-x}$ . The inverse transformation of $r_{y=-x}$ is $r_{y=x}$ , which is a reflection across the line $y = x$ .

Applying the Inverse Transformation

To find the pre-image of the point $(-4, 9)$ , we need to apply the inverse transformation $r_{y=x}$ to the point $(-4, 9)$ . This means that we need to reflect the point $(-4, 9)$ across the line $y = x$ .

Reflection Across the Line y = x

To reflect a point $(x, y)$ across the line $y = x$ , we need to swap the coordinates of the point. This means that the new x-coordinate will be the old y-coordinate, and the new y-coordinate will be the old x-coordinate.

Finding the Pre-Image Coordinates

Applying the reflection across the line $y = x$ to the point $(-4, 9)$ , we get:

New x-coordinate: Old y-coordinate = 9
New y-coordinate: Old x-coordinate = -4

Therefore, the coordinates of the pre-image of the point $(-4, 9)$ are $(9, -4)$ .

Conclusion

In conclusion, the coordinates of the pre-image of the point $(-4, 9)$ under the transformation rule $r_{y=-x}$ are $(9, -4)$ . This is because the inverse transformation of $r_{y=-x}$ is $r_{y=x}$ , which is a reflection across the line $y = x$ . By applying this reflection to the point $(-4, 9)$ , we get the pre-image coordinates $(9, -4)$ .

Answer

The correct answer is:

D. $(9, -4)$

Discussion

This problem requires a good understanding of the concept of image and pre-image under a given transformation rule. It also requires the ability to apply the inverse transformation to find the pre-image coordinates. The problem is a good example of how to use the concept of image and pre-image in geometry and coordinate geometry.

Key Terms

Image: The output of a transformation
Pre-image: The input of a transformation
Inverse transformation: A transformation that undoes the effect of another transformation
Reflection: A transformation that flips a point across a line
Q&A: Image and Pre-Image Under a Given Transformation Rule =============================================================

Frequently Asked Questions

In this article, we will answer some frequently asked questions related to the concept of image and pre-image under a given transformation rule.

Q: What is the image of a point under a transformation rule?

A: The image of a point under a transformation rule is the output of the transformation. It is the point that results from applying the transformation to the original point.

Q: What is the pre-image of a point under a transformation rule?

A: The pre-image of a point under a transformation rule is the input of the transformation. It is the original point that is transformed into the image point.

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

A: To find the pre-image of a point under a transformation rule, you need to apply the inverse transformation to the image point. The inverse transformation is a transformation that undoes the effect of the original transformation.

Q: What is the inverse transformation of a given transformation rule?

A: The inverse transformation of a given transformation rule is a transformation that undoes the effect of the original transformation. It is a transformation that, when applied to the image point, returns the original point.

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

A: To apply the inverse transformation, you need to swap the coordinates of the image point. This means that the new x-coordinate will be the old y-coordinate, and the new y-coordinate will be the old x-coordinate.

Q: What is the difference between a transformation and its inverse?

A: A transformation and its inverse are two transformations that undo each other. When you apply a transformation to a point, it produces an image point. When you apply the inverse transformation to the image point, it returns the original point.

Q: Can you give an example of a transformation and its inverse?

A: Yes, a simple example of a transformation and its inverse is the reflection across the line y = x. The transformation rule is r_{y=-x}(x, y) = (-4, 9). The inverse transformation rule is r_{y=x}(x, y) = (9, -4).

Q: How do I know if a transformation is invertible?

A: A transformation is invertible if it has an inverse transformation. This means that the transformation must be one-to-one, meaning that each input point maps to a unique output point.

Q: What is the significance of image and pre-image in geometry and coordinate geometry?

A: Image and pre-image are important concepts in geometry and coordinate geometry because they help us understand how transformations affect points and shapes. They are used to describe the effect of a transformation on a point or shape, and to find the original point or shape that was transformed.

Q: Can you give some examples of transformations and their inverses?

A: Yes, here are some examples of transformations and their inverses:

Reflection across the line y = x: r_{y=-x}(x, y) = (-4, 9), r_{y=x}(x, y) = (9, -4)
Rotation by 90 degrees: r_{90}(x, y) = (-y, x), r_{-90}(x, y) = (y, -x)
Translation by (a, b): r_{a,b}(x, y) = (x + a, y + b), r_{-a,-b}(x, y) = (x - a, y - b)

Conclusion

In conclusion, image and pre-image are important concepts in geometry and coordinate geometry. They help us understand how transformations affect points and shapes, and to find the original point or shape that was transformed. By understanding the concept of image and pre-image, we can apply transformations to points and shapes, and find the pre-image of a point under a given transformation rule.