The domain of definition of function $$f(x)=log$$ $$x2-5x-24$$ $$-x-2$$,is

Here,$$x2-5x-24$$>$$x+2$$ is equivalent to the collection of two system of inequations.Case I $$x2-5x-24$$≥$$0$$ and $$x+2$$ $$x+22$$ ⇒ x∈$$(-$$∞,$$-3$$]∪[$$8$$,∞$$),x∈[$$-2$$,∞$$)$$ and $$x2-5x-24$$>$$x2+4x+4$$ ⇒ $$9x$$<$$-28$$ ⇒ x<$$-289$$ ∴ No solution i.e., x∈ϕ ....(ii) From Eqs.(i) and (ii), we get x∈$$(-$$∞,$$-3$$]

Questions

The domain of definition of function $f (x) = \log (\sqrt{x^{2} - 5 x - 24} - x - 2),$ is

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(-∞,-3]

(-∞,-3]∪[8,∞)

-∞,-289

None of these

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detailed solution

Correct option is A

Here,x2-5x-24>x+2 is equivalent to the collection of two system of inequations.Case I x2-5x-24≥0 and x+2<0 ⇒ (x-8)(x+3)≥0 and x<-2 ⇒ x≤-3 .....(i)Case IIx2-5x-24≥0,x+2≥0 and x2-5x-24>x+22 ⇒ x∈(-∞,-3]∪[8,∞),x∈[-2,∞) and x2-5x-24>x2+4x+4 ⇒ 9x<-28 ⇒ x<-289 ∴ No solution i.e., x∈ϕ ....(ii) From Eqs.(i) and (ii), we get x∈(-∞,-3]

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