We need to prove the existence of a computable function $\backslash ($ f $\backslash )$ that satisfies the following condition for every $\backslash ($ x $\backslash )$:

$\backslash [$ f$($x$)\backslash$approx

$\backslash$begin$\{$cases$\}$

$\backslash$frac$\{$x$\}\{2\},$ & $\backslash$text$\{$if $\}$ x $\backslash$text$\{$ is even$\}\backslash \backslash$

f$($f$($f$(\backslash$frac$\{3$x$+1\}\{2\}))),$ & $\backslash$text$\{$otherwise$\}$

$\backslash$end$\{$cases$\}$

$\backslash ]$