Problem $2\mathrm{.}9\mathrm{.}4$ If f is a function from a set A to itself, we can compose f with

itself. We call the composition of f with itself k times the k$$th iterate of f$,$ and write it f

$($k$)$

$.$

$($a$)$ If f$($x$)=$ x $+2,$ what is the function f

$(3)?$

$($b$)$ If g$($x$)=$ x

$2+$ x $+1,$ what is the function g

$(3)($x$)?$

$($c$)$ If i and j are any naturals, is it always true that $($f

$($j$)$

$)$

$($k$)$

is equal to f

$($jk$)$

$?$ Why or why

not?

$($d$)$ How should we define f

$(0)?$ Why?