Question

Which of the following is true about [x], greatest integer function ?

• A.
• B.
• C.
• D.

Answer 1

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

The correct options are
A

B

C

D

(A) [x] is an integer say n, [[x]] = [n] = n, because n is an integer

(B)We can write x = [x] + {x}, where {x} denotes the fractional part function.

(For example consider 3.43 = 3 +0.43 = [3.43]+{3.43} )
[ x + n ] = $\left[\right[x]+(x)+n]$

= $\left[\right(\left[x\right]+n)+(x\left)\right]$

[x] + n is an integer, say m

$\Rightarrow$ [x+n] = [ m + {x} ]

= m

= [x] + n

Because, 0 $\le$ {x} $\le$ 1

(C) x - 1 < [x] $\le$ x

x = [x] + {x}

[x] = x - {x}

$\Rightarrow$ [x] is less than x and it will be equal to x when {x} is zero. {x} is less then 1 . So [x] = x - {x}, will be greater than x -1 .

(D) Let x be a npn - integer real number between I and I +1

$\Rightarrow$ I < x < I +1

[x] = 1 .... (1)

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

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

$\Rightarrow$ [x] = -I - 1

= - [x] - 1

( [x] = I from (1) )