Question

Which of the following options is/are true for the Greatest integer function [.]?

• A. $\left[x\right]\le x<\left[x\right]+1$
• B. $x-1<\left[x\right]\le x$
• C. $[x+m]=x+\left[m\right],$ if $m\in Z$
• D. $[x+m]=\left[x\right]+m,$ if $m\in Z$

Answer 1

Answer: (A), (B), (D)

$[x+m]=\left[x\right]+m,$ if $m\in Z$

We know that,
The real function $f:R\to R$ defined by $f\left(x\right)=\left[x\right]$ takes value less than or equal to $x$

So,
$\left[x\right]=x$ if $x$ is an integer
$\Rightarrow \left[x\right]\le x<\left[x\right]+1$
$\Rightarrow x-1<\left[x\right]\le x$
$\Rightarrow [x+m]=\left[x\right]+m,$ if $m\in Z$