Solution 18 (a).

Answer.   The fixed points of    are  .  

Solution.   Solve the equation      for  z  and get  

        

Therefore, the fixed points of    are  .  

Just for fun, we can substitute    into  the formula  .

                    ,    and  

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

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

                                        The fixed points of the mapping    are  .  

This solution is complements of the authors.

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