Question

If $[.]$ denotes the greatest integer function, then which of the following values can $\left[\right[-x\left]\right]$ hold?

• A. $-1-\left[x\right],$ if $x\notin Z$
• B. $-1-\left[x\right],$ if $x\in Z$
• C. $-x,$ if $x\notin Z$
• D. $-x,$ if $x\in Z$

Answer 1

Answer: (A), (D)

$-x,$ if $x\in Z$

From the properties of greatest integer function, $\left[\right[x\left]\right]=\left[x\right]$
Also, $\left[x\right]+[-x]=(\begin{array}{c}0,x\in Z\\ -1,x\notin Z\end{array}$

$\therefore \left(-x\right]=(\begin{array}{c}-x,x\in Z\\ -1-\left[x\right],x\notin Z\end{array}$
So, on taking greatest integer function both side, the RHS will not be affected because $\left[x\right]and-1-\left[x\right]$ are already an interger quantity. Therefore,
$\left(\right(-x\left]\right]=(\begin{array}{c}-x,x\in Z\\ -1-\left[x\right],x\notin Z\end{array}$
Hence, both options (a) and (d) are correct.