\section*{2. For ALL APPLICANTS.}
Let

$$A ( x ) = 2 x + 1 , \quad B ( x ) = 3 x + 2 .$$

(i) Show that $A ( B ( x ) ) = B ( A ( x ) )$.\\
(ii) Let $n$ be a positive integer. Determine $A ^ { n } ( x )$ where

$$A ^ { n } ( x ) = \underbrace { A ( A ( A \cdots A } _ { n \text { times } } ( x ) \cdots )$$

Put your answer in the simplest form possible.\\
A function $F ( x ) = 108 x + c$ (where $c$ is a positive integer) is produced by repeatedly applying the functions $A ( x )$ and $B ( x )$ in some order.\\
(iii) In how many different orders can $A ( x )$ and $B ( x )$ be applied to produce $F ( x )$ ? Justify your answer.\\
(iv) What are the possible values of $c$ ? Justify your answer.\\
(v) Are there positive integers $m _ { 1 } , \ldots , m _ { k } , n _ { 1 } , \ldots , n _ { k }$ such that

$$A ^ { m _ { 1 } } B ^ { n _ { 1 } } ( x ) + A ^ { m _ { 2 } } B ^ { n _ { 2 } } ( x ) + \cdots + A ^ { m _ { k } } B ^ { n _ { k } } ( x ) = 214 x + 92 \quad \text { for all } x ?$$

Justify your answer.


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