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.
Illustration: Let [x + 1] = 3 then find x.
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
[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
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.3 1 < x < 2
IIT JEE and AIEEE study material is available online free of cost at askIITians.com. Study Physics, Chemistry and Mathematics at askIITians website and be a winner. We offer numerous live online classroom courses as well for live online IIT JEE preparation - you do not need to travel anywhere any longer - just sit at your home and study for IIT JEE live online with askIITians.com