Exercise 3.  Find the image of the first quadrant   ,   under   .

Solution 3.

Answer.   The image of the first quadrant  ,  under the mapping  

  is the horizontal strip  .  

Solution.   For    in the first quadrant      we have   .    

Recall that      which can be written as  

                       and   .

Then,      implies that      implies that      in turn implies that   .  

Also      implies that      in turn implies that   .   

        Therefore,  the image of the first quadrant  ,  under the mapping  

  is the horizontal strip  .  

We are done.   

          The mapping   .

          Observe the points and their images    

                                                    

We are really done.   

Aside.   If it is important then we can include the following details.

          Here the values    ,    ,    ,        are calculated as limits.

                    ,

                    ,  

                    ,  

                    .  

This solution is complements of the authors.

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