Question

The domain of f(x) = √(x - 1/(x - 2{x}),where {x} denotes the fractional part of x, is

(A)  (−∞, 0) ∪ (0, 2]

(B)  [1, 0)

(C)  (−∞, ∞) ∼ (0, 2]

(D)  (−∞, 0)∪(0, 1]∪[2, ∞)

Answer 1

Correct option  (D)  (−∞, 0)∪(0, 1]∪[2, ∞)

Explanation :

For x - 1/x - 2{x} ≥ 0, we have the following two cases:

(i)  x ≥ 1 ⇒ x > 2{x} ⇒ x ≥ 2 ⇒ x ∈ [2, ∞)

(ii)  x ≤ 1 ⇒ x < 2{x} ⇒ x < 1, x ≠ 0 The common part is x ∈ (−∞, 0) ∪ (0, 1). Finally, x = 1 is also a part of the domain.