Question

graph (xy) with endpoints (x(-3,1)) and (y(4, - 5)) and its image after the composition.
reflection: in the line (y = x)
rotation: (90^{circ}) clockwise about the origin
write a coordinate rule for the composition.
((x,y)\to(square,square))

Explanation:

Step1: Apply reflection rule

The rule for reflection in the line $y = x$ is $(x,y)\to(y,x)$.
For point $X(-3,1)$, after reflection, $X_1=(1, - 3)$. For point $Y(4,-5)$, after reflection, $Y_1=(-5,4)$.

Step2: Apply rotation rule

The rule for a $90^{\circ}$ clock - wise rotation about the origin is $(x,y)\to(y,-x)$.
For $X_1=(1, - 3)$, after rotation, $X_2=(-3,-1)$. For $Y_1=(-5,4)$, after rotation, $Y_2=(4,5)$.

Step3: Find the coordinate rule for the composition

Let's start with a general point $(x,y)$.
First, reflection in $y = x$ gives $(y,x)$. Then, $90^{\circ}$ clock - wise rotation about the origin on $(y,x)$ gives $(x,-y)$.

Answer:

$(x,y)\to(x,-y)$