Which of the following is true about [x], greatest integer function ?
[-x] = -1 - [x] , when x is not an integer
(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 ] = [[x]+(x)+n]
= [([x]+n)+(x)]
[x] + n is an integer, say m
⇒ [x+n] = [ m + {x} ]
= m
= [x] + n
Because, 0 ≤ {x} ≤ 1
(C) x - 1 < [x] ≤ x
x = [x] + {x}
[x] = x - {x}
⇒ [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
⇒ I < x < I +1
[x] = 1 .... (1)
Now - x , will lie between - I -1 and -I
⇒ -I - 1 < - x < - I
⇒ [x] = -I - 1
= - [x] - 1
( [x] = I from (1) )