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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.
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
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]