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Greatest Integer Function

The function f(x) : R → Z defined as:

f(x) = [x] = greatest integer less than or equal to x is called the greatest integer function. The graph of a greatest integer function is shown in figure given below. The graph shows that it is increasing (not strictly) many-to-one function.

                                               many-to-one-function

Illustration: Let [x + 1] = 3 then find x.

Solution:

        From definition of greatest integer function

        3 < x + 1 < 4

        => 2 < x < 3

Note :       Any number x can be written as

                x = [x] + (x)

where [ ] denotes the integral part

and ( ) denotes the fractional part

i.e.

        [3.7]  = 3

        (3,7)  = 0.7

        [-3,7] = -4

        (-3.7) = 0.3.

Note :       0 < (x) < 1

                ∀ -2 < x < -1 => [x] = -2

                ∀ -1 < x < 0 => [x] = -1

                ∀ 0 < x < 1 => [x] = 0

                ∀ 1 < x < 2 => [x] = 1

                ∀ 2 < x < 3 => [x] =2

∀ n < x < n + 1 => [x] = n, n ε I

Examples

1.     [x + 1] = [x] + 1 ∀ x ε R                               True/False

2.     |-(x/∏)| = -1-|x/∏|, x ≠ n ∏ , n ε I                True/False

3.     If [(x) + x] = 3 then x =? where [ ] represents greatest integer function and  ( ) represents integer greater than or equal to x.

Ans.1         True

Ans.2         True

Ans.3         1 < x < 2

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