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The domain of definition of the function f x =   log x - 1 ⁡ x 2 + 4 x + 4 , is

Options

  1. A- 3 , - 1 ∪ 1 , 2
  2. B- 2 , - 1 ∪ 2 , ∞
  3. C- ∞ , - 3 ∪ - 2 , - 1 ∪ 2 , ∞
  4. D- 2 , - 1 ∪ 2 , ∞

Correct answer

C. - ∞ , - 3 ∪ - 2 , - 1 ∪ 2 , ∞

Step-by-step solution

log   x - 1 ⁡   x 2 + 4 x + 4 ≥ 0 Case 1: 0 < x - 1 < 1 i.e., 1 < x < 2 , then 0 < x 2 + 4 x + 4   ≤ 1 ⇒         x 2 + 4 x + 3   ≤ 0  &   x + 2 2 > 0 ⇒ - 3   ≤ x   ≤ - 1  &   x ≠ - 2 So, x ∈ - 2 ,   - 1 Case 2: x - 1 > 1 i.e., x > 2 x 2 + 4 x + 4   ≥ 1 ⇒ x + 1 x + 3 ≥ 0 ⇒ x ∈   - ∞ , - 3   &#1

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