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\centerline{\bigrm Life On Long Street}
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\noindent
{\bf Purpose:} In this project you will investigate compositions of
functions.
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\noindent
{\bf The Problem:} Tom lives on Long street.  Long street is straight
       and goes from El Paso to Texarkana, in other words it is infinite
       in both directions. The addresses
       along the street are given by real numbers. (Tom lives at
       $\sqrt{2}$, while his friend Mary lives at $\pi$.)  There are
       only two moving companies who will move people along Long street.
       The companies are company $f$ and company $g$.  When someone
       at address $x$ moves using company $f$ then that person is moved
       to the address $f(x)$.  When someone at address $x$ moves using
       company $g$ then that person moves to $g(x)$.  Now, the problem
       is to determine all the possible places Tom could move by using
       the companies $f$ and $g$.  He can use each company
       as many times as he
       wishes and he can use them in any order he wishes.
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\noindent
{\bf Procedure:}  You are to follow the steps below.  You do not need to
do the steps in exactly the order indicated as long as you cover
everything you are asked to do.
After completing the outline below you should write your
results and calculations {\bf neatly}.
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\noindent
{\bf Project Outline:}
\item{1.}Consider the functions $f(x)=1-x$ and $g(x)={1\over x}$.
  \itemitem{a.}Find all possible functions you can obtain by
               taking repeated compositions of $f$ and $g$.
  \itemitem{b.}Prove that you get the functions
               you say and no more.
  \itemitem{c.}Based on these computations determine
               the possible addresses Tom could have if he moves.
  \itemitem{d.}If company $g$ attempts to move someone at address 0 then
               that person moves to Arkansas.  What are the possible
               addresses one could have so that moving with $f$ and $g$
               could make that person move to Arkansas?

\item{2.}Now consider the functions $f(x)=2x$ and $g(x)={1\over x}$.
         There are an infinite number of different functions you can get
         by repeatedly composing these functions.  Find a general form
         for all the functions you can get by taking repeated
         compositions of these two functions and prove that you get
         exactly the functions you claim.  Based on these computations
         determine where Tom can move.
\item{3.}Notice that both functions $f$ and $g$ in Part~1 could be
         written in the form ${ax+b\over cx+d}$ and both have the
         property that when you compose the function with itself you get
         the identity function $i(x)=x$.
  \itemitem{a.}Determine all values for $a$, $b$, $c$ and
               $d$ which make $f(x)={ax+b\over cx+d}=x$.
               (That is, find equations
               which these numbers satisfy exactly when $f(x)=x$ as long
               as $x\not= -{d\over c}$.)
               Prove your answer.
  \itemitem{b}Find all functions of the form
              $f(x)={ax+b\over cx+d}$ with the property that
              $f\circ f(x)=x$ for all values of $x$ in the domain of
              $f\circ f$. (Be careful, consider all the
              cases.)

\bye
