Question

If $f\left(x\right)=3[x+1]+2[x+2]+[x-5],$ then the value of $f\left(2.999\right)$ is
(where [.] denotes the greatest integer function)

Answer 1

$f\left(x\right)=3[x+1]+2[x+2]+[x-5]$
$\Rightarrow f\left(x\right)=3(\left[x\right]+1)+2(\left[x\right]+2)+\left[x\right]-5$
$(\because [x+I]=\left[x\right]+I,\forall I\in Z)$
$\Rightarrow f\left(x\right)=3\left[x\right]+3+2\left[x\right]+4+\left[x\right]-5$
$\Rightarrow f\left(x\right)=6\left[x\right]+2$
$\therefore f\left(2.999\right)=6\left[2.999\right]+2$
$=6\times 2+2$
$=14$