Question

Which of the following statements are true about [x], greater integer Function?

1.$x-1)<[x]\le x$

2.$[-x]=1-\left[x\right]$, when x is NOT an integer

• A. Only 1
• B. Only 2
• C. Both 1 and 2
• D. None of these

Answer 1

The correct option is A

Only 1

1.$x-1<\left[x\right]\le x$

We know, $x=\left[x\right]+\left\{x\right\}$

Where [.] and {.} are greatest integer and fractional part of function respectively.

$\left[x\right]=x-\left\{x\right\}$

$\Rightarrow \left[x\right]=x-\left\{x\right\}$

$\left[x\right]$ is less tahn x, it will be equal to x when {x} is zero.

Fractional part of function is less tahn 1.so,

$\left[x\right]=x-\left\{x\right\})$ will be greater than $x-1$

$x-1<\left[x\right]\le x$

statement 1 is correct.

2.Let x be a non integer real number between I and I+1

$\Rightarrow I<x<I+1$

$\left[x\right]=I$

Now,-x will be between -I -1 and -I

$\Rightarrow -I-1<-x<-I$

$\left[x\right]=-I-1$

$[-x]=-\left[x\right]-1$

$[-x]=-1-\left[x\right]$

Statement 2 is wrong.

Only statement 1 is correct.