Question

Let $f$ be a function on non negative integers defined as follows

$f\left(2n\right)=f\left(f\left(n\right)\right)$
$f(2n+1)=f\left(2n\right)+1:$
-If $f\left(0\right)=0$, find $f\left(n\right)$ for every $n$.

Answer 1

$f\left(0\right)=0$
$\Rightarrow f(2\times 0+1)=f(2\times 0)+1$
$f\left(1\right)=f\left(0\right)+1\Rightarrow f\left(1\right)=1$
Now, $f(2\times 1)=f\left(f\right(12\left)\right)\Rightarrow f\left(2\right)=f\left(1\right)=1$
$\Rightarrow f(2\times 1+1)=f(2\times 1)+1$
$f\left(3\right)=f\left(2\right)+1\Rightarrow f\left(3\right)=1+1\Rightarrow f\left(3\right)=2$
$\Rightarrow f(2\times 2)=f\left(f\right(2\left)\right)=f\left(1\right)=1$
$\Rightarrow f(2\times 2-1)=f\left(4\right)+1$
$=1+1=2$
$f\left(5\right)=2$
$\Rightarrow f\left(1\right)=f\left(2\right)=f\left(4\right)=.......=1$
$f\left(3\right)=f\left(5\right)=f\left(7\right)=......=2$