Question

Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.

Options

• –3

• 1

• –2

• 8

Answer 1

Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is –2.

Explanation:

f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)`

⇒ `sqrt((|[x]| - 2)/(|[x]| - 3)) ≥ 0 ∩ |[x]| -3 ≠ 0`

Let t = |[x]|, t > 0

⇒ `sqrt((t - 2)/(t - 3)) ≥ 0` ⇒ `(t - 2)/(t - 3) ≥ 0`

⇒ t ∈ (–∞, 2] ∪ (3, ∞) ∩ t > 0

⇒ |[x]| ∈ [0, 2] ∪ (3, ∞)

⇒ [x] ∈ (–∞, –3) ∪ [–2, 2] ∪ (3, ∞)

⇒ x ∈ (–∞, –3) ∪ [–2, 3) ∪ [4, ∞)

So, a = –3, b = –2, c = 3

So, a + b + c = –3 – 2 + 3 = –2