Let's start with the smallest number. (You can begin at any number)
$f(1) = f(6) = f(3) = f(8) = f(4) = f(2) = \color{red}{f(1)}... $
$f(2)=\color{red}{f(1)}$
$f(3)=\color{red}{f(1)}$
$f(4)=\color{red}{f(1)}$
$f(5) = f(10) = \color{blue}{f(5)}... $
$f(6)=\color{red}{f(1)}$
$f(7) = f(12) = f(6) =\color{red}{f(1)}$
$f(8)=\color{red}{f(1)}$
$f(9) = f(14) = f(7)=\color{red}{f(1)}$
$f(10)=\color{blue}{f(5)}$
$f(11) = f(16) = f(8)=\color{red}{f(1)}$
... so on.
We observe that all the multiples of 5 will have the value of $f(5)$ and every other number will converge to $f(1)$ ultimately.
Let's assume $f(1) = i_1$ and $f(5) = i_2$
$R=\{i∣∃j:f(j)=i\} $
Hence, there are 2 such $i$'s.