The range of the function f(x)

f(x) = [x] − x

Since x ≥ [x]

Every number is greater than or equal to its greatest integral value

i.e x – [x] = {x} fractional part of x. 

∴ [x] − x = −{x} fraction part only. 

also [x] = x for integral value of x 

hence, for non-integral values  f(x) = −{x} ∈">∈∈ (−1, 0) 

and for integral values  f(x) = 0

Hence, Range of f(x) is (−1, 0].