question_answer

The domain of the function\[{{\cos }^{-1}}({{\log }_{2}}({{x}^{2}}+5x+8))\]is

A)  \[[2,\text{ }3]\]                               

B)  \[[-2,\text{ }2]\]

C)  \[[3,\text{ }1]\]               

D)        \[(-2,-2)\]

E)  \[[-3,-2]\]

Correct Answer: E

Solution :

Function\[{{\cos }^{-1}}({{\log }_{2}}({{x}^{2}}+5x+8))\] The function should exist when \[-1\le lo{{g}_{2}}\text{(}{{x}^{2}}+5x+8)\le 1\] We take only, \[lo{{g}_{2}}\text{(}{{x}^{2}}+5x+8)\le 1\] \[\Rightarrow \]               \[{{x}^{2}}+5x+8\le {{(2)}^{1}}\] \[\{\because -1\le {{\log }_{2}}({{x}^{2}}+5x+8)\] given imaginary values} \[\Rightarrow \]               \[{{x}^{2}}+5x+6\le 0\] \[\Rightarrow \]               \[(x+2)(x+3)\le 0\] \[\Rightarrow \]               \[x\in [-3,-2]\] Hence, the domain of the function is \[-3\le x\le -2\]