Explain the Greatest integer function - Maths - Relations and Functions

Question

Explain the Greatest integer function

Brijendra Pal · Accepted Answer

Hi,

The function f : R → R defined by

f (x) = [x], x ∈ R, assuming the value of the greatest integer less then or equal to x is called greatest integer function.

Example : The greatest integer less than or equal to 3.9 is 3

∴ [3.9] = 3

Now we can apply the same in question as:

The function f : R → R defined by

f ( x ) = [ x ] ∀ x ∈ R is called the greatest integer function.

For n ≤ x < n + 1, [ x ] = n, where n is an integer.

Example:

[ 2.57 ] = 2

[– 3.87] = – 4

Domain of f (x) = R (Set of all real numbers)

Range of f (x) = Z (Set of all integers)

Find domain and range of $f (x) = \frac{1}{\sqrt{x - [x]}}$

We know that, 0 ≤ x – [ x ] < 1 ∀ x ∈ R

x – [ x ] = 0 ∀ x ∈ Z.

∴ 0 < x – [ x ] < 1 ∀ x ∈ R – Z

$\Rightarrow f (x) = \frac{1}{\sqrt{x - [x]}} exists for all x \in R - Z$

Domain of f (x) = R – Z.

0 < x – [ x ] < 1 ∀ x ∈ R – Z

$\Rightarrow 0 < \sqrt{x - [x]} < 1 \forall x \in R - Z \Rightarrow 1 < \frac{1}{\sqrt{x - [x]}} < \infty \forall x \in R - Z \Rightarrow 1 < f (x) < \infty \forall x \in R - Z$

Range of f (x) = (1, ∞)