To determine the pre-image coordinates, we need to understand the transformation rule given: [tex]\( r_y = -x(x, y) \)[/tex]. This rule implies the following steps for transforming a point [tex]\((x, y)\)[/tex] into its image:
1. Swap the coordinates [tex]\( (x, y) \rightarrow (y, x) \)[/tex].
2. Negate the new [tex]\(x\)[/tex] coordinate: [tex]\((y, x) \rightarrow (-x, y) \)[/tex].
Given the transformed coordinates are [tex]\((-4, 9)\)[/tex], we need to work backward through the rule to find the original coordinates.
Step-by-step reconstruction:
1. We know the transformed point is [tex]\((-4, 9)\)[/tex].
2. Reverse the negation and get the coordinate that was negated and swapped.
To find the original [tex]\(x\)[/tex]:
- The negated and swapped value of [tex]\(x\)[/tex] is -4.
To find the original [tex]\(y\)[/tex]:
- The coordinate for [tex]\(y\)[/tex] was not affected by negation, so it remains 9.
Hence, we reverse these transformations to find:
1. Swap back the coordinates: The 'y' here would be the original 'x', and the 'x' here would be the original 'y'.
2. Considering the negation, the original 'x' would have been [tex]\(9\)[/tex] and the original 'y' would have been [tex]\(-4\)[/tex].
Thus, the original pre-image coordinates are [tex]\((9, -4)\)[/tex].
The correct choice is:
[tex]\[
(9, -4)
\][/tex]
