Mathematics

10. Fractional Part f(x) = {x}  Every number 'x' can be written as the sum of its integer and fractional parts. x = [x] + {x} Therefore fractional part, {x} = x - [x] For example, 4.7 = 4 + 0.7 or, 0.7 = 4.7 - 4 i.e. fractional part = Number - Integral Part x = 4.7 , [x] = [4.7] = 4 and {x} = {4.7} = 0.7 Consider, {-4.7} {-4.7} = -4.7 - [-4.7] = -4.7 - (-5)  or, {-4.7} = 0.3 if the number is integer, its fractional part is obviously zero. {6} = 6 - [6] = 6 - 6 = 0 Note- fractional part is always positive and it never becomes 1. in other words, fractional part of a number is the difference between the number and its integral part. Remember,  i.e. fractional part is always less than 1 and greater than or equal to zero. When x will be an integer fractional part will become zero i.e. integer has no fractional part. How to draw the graph, note this, , if  because the fractional part is equal to the number if number is between

1. Domain of    where [ ] denotes greatest integer function. We Know that , Domain is the Possible point of   for which we can get finite and real value of  , to know more [ click here ] Consider, for finite and real value,  ,   should not be negative, therefore,  ( denominator cant be equal to zero, so it is strictly greater than zero) or,  but we know that ,   ( to know more about Greatest Integer Function click here  ) => for no value of x, f(x) will be real and finite. so domain of f(x) = { } =  we can also see this from graph, Here, we can see that for a point on x - axis or for a value of x , graph of y = x is above the graph of y = [x] and both the graph touch only on integer points.            You can check this by taking a point on x- axis and draw a vertical line through this in upward direction , you can see that the line first cut the graph of y = [x] and then cut the graph of y = x , this shows for any value of x ,  2.Domain of 

We have understand some properties and graph of greatest integer function on my previous post [ click here ] Understand this, [2] = 2 [2.09] = 2 [2.8] = 2 [2.98] = 2 [3] = 3 i.e. for   , [x] = 2 it can also see from the graph for  , y = 2 1. Consider, [x] = 5 what is the solution of this equation ?? [x] = 5 =>   or  and, [x] = -3  then ,   or  most funny one, [x] = 0 just don't think x = 0 if [x] = 0, then   or  So, if   and I is an integer, then   or  2.Consider, [x] = 5.6 (No Solution) why??? because greatest integer function gives only integer values. then there is no solution and for no value of x the above equation satisfied. so if   and n is not an integer, then there is no solution. 3. Consider, [x] > 5 we know that, for   or  , [x] = 5 so if [x] > 5, then  in general, , then   where I is an integer   4. Consider, [x] > 5.3 we know that,  for   or  , [x] = 5 so if [x] > 5.3, then  in general