Chapter 14, Problem 3CR

a.

To determine

To find : the image and preimage of $\left(3,3\right)$

Answer to Problem 3CR

The image is $\left(6,1\right)$ and the preimage is $\left(\frac{3}{2},5\right)$ .

Explanation of Solution

Given information : If $S:\left(x,y\right)\to \left(2x,y-2\right)$ .

Calculation :

Image of $\left(3,3\right)$ is given as:

  $\begin{array}{l}S:\left(x,y\right)\to \left(2x,y-2\right)\\ S:\left(3,3\right)\to \left(2\left(3\right),3-2\right)\\ S:\left(3,3\right)\to \left(6,1\right)\end{array}$

Preimage of $\left(3,3\right)$ is given as:

  $\begin{array}{l}\left(2x,y-2\right)=\left(3,3\right)\\ x=\frac{3}{2},y=5\\ \left(3,3\right)=\left(\frac{3}{2},5\right)\end{array}$

b.

To determine

To check : that $S$ is isometry.

Answer to Problem 3CR

  $S$ is not an isometry.

Explanation of Solution

Given information : If $S:\left(x,y\right)\to \left(2x,y-2\right)$ .

Image and preimage is not equal.

Thus, $S$ is not an isometry.