Question

what are the coordinates of the vertices of the pre - image given? $r_{y=-x}circ t_{1,-2}(x,y)$
a
b
c (3,4)
d
(-1,-5)
(2,-6)
(1,5)
(2,6)

Explanation:

Step1: Analyze translation rule

The transformation $T_{1,-2}(x,y)=(x + 1,y - 2)$ means a translation 1 unit to the right and 2 units down. The transformation $r_{y=-x}(x,y)=(-y,-x)$ is a reflection across the line $y=-x$.
Let the pre - image coordinates be $(x,y)$. After translation $T_{1,-2}$, the coordinates become $(x + 1,y - 2)$. Then after reflection $r_{y=-x}$, the new coordinates $(x',y')$ are given by $x'=-(y - 2)$ and $y'=-(x + 1)$.

Step2: For point C''(3,4)

Let's work backwards. First, reverse the reflection. If $(x',y')=(3,4)$ after reflection across $y =-x$, then before reflection (after translation), the coordinates $(x_1,y_1)$ satisfy $x_1=-4$ and $y_1=-3$. Then, to reverse the translation, we set $x_1=x + 1$ and $y_1=y - 2$. So $x=x_1 - 1=-4 - 1=-5$ and $y=y_1+2=-3 + 2=-1$.
Let's assume we know the general rules for inverse of transformations. For a point $(x,y)$ in the pre - image, after $T_{1,-2}$ we have $(x_1,y_1)=(x + 1,y - 2)$ and after $r_{y=-x}$ we have $(x_2,y_2)=(-(y - 2),-(x + 1))$.
For point with final image coordinates $(a,b)$:
Reverse reflection: If $(a,b)$ is the image after $r_{y=-x}$, the point before reflection (after translation) is $(-b,-a)$. Then reverse translation: $x=-b - 1$ and $y=-a + 2$.
For point C''(3,4):
Reverse reflection gives $(-4,-3)$. Reverse translation: $x=-4 - 1=-5,y=-3 + 2=-1$.
Let's take another approach.
The composition of transformations $r_{y=-x}\circ T_{1,-2}(x,y)=(-(y - 2),-(x + 1))$.
If the image point is $(x_{image},y_{image})$, then to find the pre - image $(x_{pre - image},y_{pre - image})$ we solve the system:
$x_{image}=-(y_{pre - image}-2)$ and $y_{image}=-(x_{pre - image}+1)$.
For point C''(3,4):
$3=-(y_{pre - image}-2)\Rightarrow y_{pre - image}-2=-3\Rightarrow y_{pre - image}=-1$
$4=-(x_{pre - image}+1)\Rightarrow x_{pre - image}+1=-4\Rightarrow x_{pre - image}=-5$
For point with final coordinates $(x_f,y_f)$:
We have the inverse transformation equations:
$x=-y_f - 1$ and $y=-x_f+2$

Answer:

For point C''(3,4), the pre - image coordinates are $(-5,-1)$. Since the options are not fully visible for A, B and D, assume we are just focusing on C. If we had to work with all points, we would apply the same reverse - transformation process for each point's final coordinates. If we assume the question is asking for the pre - image of C(3,4) among the given options, none of the provided options match the correct pre - image coordinates we calculated. There may be an error in the problem setup or the provided options. If we had to make a choice based on the work shown and the available options, we note that our calculated pre - image for C(3,4) is not among them.