Exercise 2.  Find the image of the vertical strip    under the mapping   .

Solution 2.

Answer.   The image of the vertical strip      under      is the right half-plane   .

Hint.   Extend the results in Example 10.12.

Short Solution.   In Example 10.12 we showed that    can be written as the composition    where  

                         and    .  

Recall that the product    rotates the plane and magnifies by a factor of  .  

Hence   maps the vertical strip    

one-to-one and onto the upper half-plane .  

Next, consider the mapping   .  

Then the three points  ,  ,    give the upper half plane    a left orientation,

and their image points  ,  ,    give the right half plane    a left orientation.  

Therefore, the image of      under      is the right half-plane   .

We might be done.   

Solution. Method I.   In Example 10.12 we showed that the mapping    can be written as the composition  

                     ,    

where   

                    ,    and   

                    .

        Recall that the product    rotates the plane and magnifies by a factor of  .  

For the mapping  ,   also recall that   

                    .  

Then      implies that      implies that      implies that   .   

Also, recall that   .

Then      implies that      implies that      implies that   .

        Hence, the image of   ,   under      is

                    ,

which is the upper half-plane.

        Now find the inverse transformation for   .

Apply equations (10-13)  and  (10-14)  in the form

(10-13)             ,

(10-14)             .

Here we have    and   .  

Then  

                        

Thus, the inverse transformation is   .  

Then get  

                       

Here we have

                        and    .  

Also,      implies       implies      implies   .

        Hence, the image of   ,   under      is

                     ,  which is the right half-plane.

        Therefore, the image of      under      is the right half-plane   .

We are done.   

Aside.  We can look at some graphs of the mapping  .

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                                              

We are really done.   

Solution. Method II.   In Example 10.12 we showed that the mapping    can be written as the composition  

                     ,    

where   

                    ,    and   

                    .

        Recall that the product    rotates the plane and magnifies by a factor of  .  

For the mapping  ,   also recall that   

                    .  

Then      implies that      implies that      implies that   .   

Also, recall that   .

Then      implies that      implies that      implies that   .

        Hence, the image of   ,   under      is

                    ,    which is the upper half-plane.

        The upper half-plane    can be given a left orientation with the points  .   

Then      maps      onto  ,

which is a left orientation for the right half plane  .

        Hence, the image of   ,   under      is the right half-plane   .

        Therefore, the image of      under      is the right half-plane   .

We are really really done.   

Aside.  We can look at some graphs of the mapping  .

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                                              

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell