To determine the coordinates of the pre-image given the transformation rule [tex]\( r_{y=-x}(x, y) \rightarrow (-4, 9) \)[/tex], let's analyze how this transformation works.
The transformation [tex]\( r_{y=-x} \)[/tex] reflects a point [tex]\((x, y)\)[/tex] over the line [tex]\( y = -x \)[/tex]. When a point [tex]\((x, y)\)[/tex] is reflected over this line, the coordinates change according to the following rule:
[tex]\[ (x, y) \rightarrow (-y, -x) \][/tex]
We are given that the image of a point under this transformation is [tex]\( (-4, 9) \)[/tex]. Therefore, we need to find the original coordinates [tex]\((x, y)\)[/tex] such that:
[tex]\[ (-y, -x) = (-4, 9) \][/tex]
From this equation, we can set up the following system of equations:
[tex]\[ -y = -4 \][/tex]
[tex]\[ -x = 9 \][/tex]
Solving these, we get:
[tex]\[ y = 4 \][/tex]
[tex]\[ x = -9 \][/tex]
Therefore, the coordinates of the pre-image are:
[tex]\[ (-9, 4) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{(-9, 4)} \][/tex]
