To find the coordinates of the pre-image of the point under the transformation rule [tex]\(r_{y=-x}\)[/tex], let's examine the given rule.
The rule [tex]\(r_{y=-x}(x, y) \rightarrow (-y, -x)\)[/tex] indicates that under this transformation:
- The x-coordinate becomes the negation of the y-coordinate.
- The y-coordinate becomes the negation of the x-coordinate.
We are given that the point transforms to [tex]\( (-4, 9) \)[/tex].
Now, reverse the transformation to find the pre-image:
- The x-coordinate of the pre-image will be the negated y-coordinate of the image.
- The y-coordinate of the pre-image will be the negated x-coordinate of the image.
Given the image [tex]\( (-4, 9) \)[/tex]:
- The x-coordinate of the pre-image will be [tex]\(-9\)[/tex] (negation of 9).
- The y-coordinate of the pre-image will be [tex]\(4\)[/tex] (negation of -4).
Thus, the coordinates of the pre-image are [tex]\( (-9, 4) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{(-9, 4)} \][/tex]
