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Please solve rd sharma class 12 chapter inverse trigonometric functions exercise 3.2 question 1 maths textbook solution

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Answer: Domain of f(x) is \left [ -\sqrt{5},-\sqrt{3} \right ]\cup \left [ \sqrt{3},\sqrt{5} \right ]

Hint: Domain of \cos ^{-1}\left ( x \right ) lies in the interval [-1, 1].

Given: f(x)=\cos ^{-1}\left ( x^{2}-4 \right )

                The domain of  \cos ^{-1}\left ( x \right ) is [-1, 1]

                \therefore -1\leq x^{2}-4\leq 1

                \Rightarrow -1+4\leq x^{2}-4+4\leq 1+4

                \Rightarrow 3\leq x^{2}\leq 5

                \Rightarrow \pm \sqrt{3}\leq x\leq \pm \sqrt{5}

                \Rightarrow x\in \left [ -\sqrt{5},-\sqrt{3} \right ]\cup \left [ \sqrt{3},\sqrt{5} \right ]

Hence, the domain of f(x) is \left [ -\sqrt{5},-\sqrt{3} \right ]\cup \left [ \sqrt{3},\sqrt{5} \right ]

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