Let $$f ( x ) = x + 1 \quad \text { and } \quad g ( x ) = 2 x$$ We will, for example, write $f g$ to denote the function "perform $g$ then perform $f$ " so that $$f g ( x ) = f ( g ( x ) ) = 2 x + 1$$ If $i \geqslant 0$ is an integer we will, for example, write $f ^ { i }$ to denote the function which performs $f i$ times, so that $$f ^ { i } ( x ) = \underbrace { f f f \cdots f } _ { i \text { times } } ( x ) = x + i .$$ (i) Show that $$f ^ { 2 } g ( x ) = g f ( x )$$ (ii) Note that $$g f ^ { 2 } g ( x ) = 4 x + 4$$ Find all the other ways of combining $f$ and $g$ that result in the function $4 x + 4$. (iii) Let $i , j , k \geqslant 0$ be integers. Determine the function $$f ^ { i } g f ^ { j } g f ^ { k } ( x )$$ (iv) Let $m \geqslant 0$ be an integer. How many different ways of combining the functions $f$ and $g$ are there that result in the function $4 x + 4 m$ ?
(i) [2 marks] Note that
\section*{2. For ALL APPLICANTS.}
Let
$$f ( x ) = x + 1 \quad \text { and } \quad g ( x ) = 2 x$$
We will, for example, write $f g$ to denote the function "perform $g$ then perform $f$ " so that
$$f g ( x ) = f ( g ( x ) ) = 2 x + 1$$
If $i \geqslant 0$ is an integer we will, for example, write $f ^ { i }$ to denote the function which performs $f i$ times, so that
$$f ^ { i } ( x ) = \underbrace { f f f \cdots f } _ { i \text { times } } ( x ) = x + i .$$
(i) Show that
$$f ^ { 2 } g ( x ) = g f ( x )$$
(ii) Note that
$$g f ^ { 2 } g ( x ) = 4 x + 4$$
Find all the other ways of combining $f$ and $g$ that result in the function $4 x + 4$.\\
(iii) Let $i , j , k \geqslant 0$ be integers. Determine the function
$$f ^ { i } g f ^ { j } g f ^ { k } ( x )$$
(iv) Let $m \geqslant 0$ be an integer. How many different ways of combining the functions $f$ and $g$ are there that result in the function $4 x + 4 m$ ?