If f is a function defined on ( 0 , 1 ) by f ( x ) = min { x - [ x ] , - x - [ x ] } , then ( f o f o f o f…

Given that, $f$ is defined on $\left(0, 1\right)$ by $f\left(x\right)=\min \{x-[x],-x-[x\left]\right\}$

We know that Greatest Integer Function is a function that gives the greatest integer less than or equal to the number.

If $x\in \left(0, 1\right)$, then $\left[x\right]=0$

Also, for $x\in \left(0, 1\right)$, since $x$ is positive, we have $x>-x$

$\Rightarrow f\left(x\right)=x$

Now, $\left(fofofof\right)\left(x\right)=\left(fofof\right)f\left(x\right)$

$=\left(fofof\right)\left(x\right)$

$=\left(fof\right)f\left(x\right)$

$=\left(fof\right)\left(x\right)$

$=f\left(f\right(x\left)\right)=x$

Hence, $\left(fofofof\right)\left(x\right)=x$.