The function f x = s i n - 1 2 x - x 2 + 2 - 1 x + 1 x 2 is defined in the interval (where · is the greatest integer function)

Question

The function f x = s i n - 1 2 x - x 2 + 2 - 1 x + 1 x 2 is defined in the interval (where · is the greatest integer function), x ∈ 1 - 2 , 1, x ∈ 1,1 + 2, x ∈ 1 - 2 , 1 + 2, x ∈ 1 - 2 , 2

MARKS App · Accepted Answer

(i) - 1 ≤ 2 x - x 2 ≤ 1 (for s i n - 1 to be defined) ⇒ - 1 ≤ x 2 - 2 x ≤ 1 i.e. x 2 - 2 x + 1 ≥ 0 and x 2 - 2 x - 1 ≤ 0 x - 1 2 ≥ 0 and x - 1 2 - 2 2 ≤ 0 x ∈ R and x - 1 - 2 x - 1 + 2 ≤ 0 ⇒ x ∈ 1 - 2 , 1 + 2 … 1 (ii) 2 - 1 x ≥ 0 ⇒ 1 x ≤ 2 ⇒ x ≥ 1 2 ⇒ x ∈ - ∞ , - 1 2 ∪ 1 2 , ∞ … 2 (iii) x 2 ≠ 0 ⇒ x 2 ∉ 0,1 ⇒ x ∉ - 1,1 ⇒ x ∈ - ∞ , - 1 ∪ 1 , ∞ … 3 Hence, 1 ∩ 2 ∩ 3 ⇒ x ∈ 1,1 + 2

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