MARVELOUS MARK'S FUNCTION MACHINES

Mark has set up a series of three function machines that he claims will surprise you.

1. Try a few numbers. Are you surprised by your results?

   Hint (a):

   Your output should be the same number as your input.

2. Carrie claims that she was not surprised by her results. She also says that she can show why the sequence of machines does what it does by simply dropping in a variable and writing out step-by-step what happens inside each machine. Try it. (Use something like $c$ or $m$.) Be sure to show all of the steps.

   Example (b):

   Input: $c$

   Step 1(b):

   function 1: $(c)−5=c−5$

   Step 2(b):

   function 2: 
   $\frac{6(\textit{c}-5)+8}{2}=\frac{6\textit{c}-30+8}{2}$ 

   Step 3(b):

   Simplify the output of function two, and then substitute it for $x$ in function three.