Question

The range of the function f(x) = [x] − x is __________ .

Answer 1

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} $\in$ (−1, 0)
and for integral values f(x) = 0
Hence, Range of f(x) is (−1, 0].