Let $$\left[ x \right]$$ denote the greatest integer $$ \leqslant x.$$  If $$f\left( x \right) = \left[ x \right]$$   and $$g\left( x \right) = \left| x \right|,$$   then the value of $$f\left( {g\left( {\frac{8}{5}} \right)} \right) - g\left( {f\left( { - \frac{8}{5}} \right)} \right)$$      is :

Solution :
$$\eqalign{ & {\text{Given that, }}f\left( x \right) = \left[ x \right]{\text{ and }}g\left( x \right) = \left| x \right| \cr & {\text{Now, }}f\left( {g\left( {\frac{8}{5}} \right)} \right) = g\left( {\frac{8}{5}} \right) = \left[ {\frac{8}{5}} \right] = 1 \cr & {\text{and }}g\left( {f\left( { - \frac{8}{5}} \right)} \right) = g\left( {\left[ { - \frac{8}{5}} \right]} \right) = g\left( { - 2} \right) = \left| { - 2} \right| = 2 \cr & \therefore \,f\left( {g\left( {\frac{8}{5}} \right)} \right) - g\left( {f\left( { - \frac{8}{5}} \right)} \right) = 1 - 2 = - 1 \cr} $$