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## Sagot :

The correct transformation sequence that accomplishes this mapping is the third one:

[tex]\[ R_{0,270}^{\circ} \circ r_{y-2 \text{ais}}(x, y) \][/tex]

This sequence involves two transformations:

1.

**Rotation [tex]\( R_{0,270^{\circ}} \)[/tex]**: This specifies a rotation about the origin (0,0) by 270 degrees counterclockwise. This changes the orientation of the shape.

2.

**Reflection [tex]\( r_{y-2 \text{ais}}(x, y) \)[/tex]**: This specifies a reflection across a specific line or axis. Here "y-2 ais" appears to be reflecting over the y-axis with some horizontal translation involved.

In the context of the steps, the pre-image PQRS undergoes the following transformations:

1.

**Rotation by 270 degrees around the origin**: This transformation will realign the points of PQRS to a new orientation.

2.

**Reflection**: This takes the already rotated shape and reflects it across the specified line.

Hence, only one of the options fits these criteria:

[tex]\[ R_{0,270}^{\circ} \circ r_{y-2 \text{ais}}(x, y) \][/tex]

So, the correct rule for the composition of transformations that maps the pre-image PQRS to the image [tex]\( P'Q'R'S' \)[/tex] is described by the third option:

[tex]\[ \boxed{3} \][/tex]