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Question

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


A

[[x]]=[x]

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B

[x + n ] = [x] + n , n I

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C

x - 1 < [x] x

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D

[-x] = -1 - [x] , when x is not an integer

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Solution

The correct option is D

[-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) )



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