Find the domain of definition of f(x)

Question:

Find the domain of definition of $f(x)=\cos ^{-1}\left(x^{2}-4\right)$.

Solution:

For $\cos ^{-1}\left(x^{2}-4\right)$ to be defined

$-1 \leq x^{2}-4 \leq 1 \Rightarrow 3 \leq x^{2} \leq 5 \Rightarrow x \in[-\sqrt{5},-\sqrt{3}] \cup[\sqrt{3}, \sqrt{5}]$

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

 

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