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Fractional Part Function

 

Definition of Fractional Part Function

If x is any number then x can be broken down into two (i.e) integral part and fractional part.

x → number

[x]→ integral part of x

{x} → fractional part of x

So any number x can be written in the form of x= [x] + {x}

Ex: 1.28=1+0.28, where 1 is the integral part and 0.28 is the Fractional Part

Therefore, the fractional part function is got by bringing the integral part a function towards LHS.

We get, {x} = x - [x]

It is also known as the decimal part function.

Graph of Fractional Part Function

image

Properties of Function Part Function

image2

Solved Examples:

Example 1: Find the fractional part function for given values of x:

(i) 5.26 (ii) -2.91 (iii) 0 (iv) 4

Solution: using the formula of the fractional part function

{x} = x - [x]

We get,

(i) {5.26} = 5.26 - 5 = 0.26

(ii){-2.91 } = -2.91 – (-3) = -2.91 + 3 = 0.09

(iii) {0} = 0 - 0 = 0

(iv){4} = 4 - 4 = 0

Answer: (i) {5.26 } = 0.26,(ii) {-2.91 } = 0.09,(iii) {0} = 0,(iv) {4} = 0

Example 2 : Find the domain of given function f(y) = 1/({y})

Solution:

To find the domain of a reciprocal of fractional part function, we want to observe places where the denominator is zero.

We know from the property of fractional part function that

image3

Hence, for f(y) = 1/({y}) to be defined, y should not be an integer.

Then the domain of f(y) = 1/({y}) consists of all real numbers excluding the integers.

(i.e) Domain = R – Z

Example:3 Solve 4{x} = x + [x]

Solution

We know, x = [x] + {x}.

4{x} = [x] + {x} + [x]

image4

 

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