Let \[f(x)=x+1\qquad\text{and}\qquad g(x)=2x.\] We will, for example, write \(fg\) to denote the function “perform \(g\) then perform \(f\)” so that \[fg(x)=f(g(x))=2x+1.\] If \(i\geq 0\) is an integer we will, for example, write \(f^i\) to denote the function which performs \(f\) \(i\) times, so that \[f^i(x)=\underbrace{fff......f}_\text{$i\,\text{times}$}(x)=x+i.\]
Show that \[f^2g(x)=gf(x).\]
Note that \[gf^2g(x)=4x+4.\] Find all the other ways of combining \(f\) and \(g\) that result in the function \(4x+4\).
Let \(i,j,k\geq0\) be integers. Determine the function \[f^igf^jgf^k(x).\]
Let \(m\geq0\) be an integer. How many different ways of combining the functions \(f\) and \(g\) are there that result in the function \(4x+4m\)?
