Greatest Integer Function IIT JEE\

I want to know the properties of Greatest Integer Function for preparation of IIT Joint Entrance Exam JEE Exam?

[Rajkumar Agarwal]

Ok, here I am telling you the properties of Greatest Integer Function for IIT Joint Entrance Exam JEE Exam.

Properties of Greatest Integer Function

[x] = n (n ∈ I) ⇒ x ∈ [n, n +1)
x – 1 < [x] ≤ x
[–x] + [x] = 78047.png
[x] ≥ n ⇒ x ≥ n, n ∈ I
[x] ≤ n ⇒ x < n + 1, n ∈ I
[x] > n ⇒ x ≥ n + 1, n ∈ I

For example,

[x] ≥ 2 ⇒ x ∈ [2, ∞)

[x] > 3 ⇒ [x] ≥ 4 ⇒ x ∈ [4, ∞)

[x] ≤ 3 ⇒ x ∈ (–∞, 4)

Fractional part function

y = f(x) = {x} = x – [x]

Domain → R; Range → [0, 1); Period → 1;

[x + y] = [x] + [y], if 0 ≤ {x} + {y} < 1

[x + y] = [x] + [y] + 1 , 1 ≤ {x} + {y} < 2

{x} + {–x} = 0 if x ∈ I

{x} + {–x} = 1 if x ∉ I

Ex-

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