Find the domain of f(x)= ^ -1 (x^2-4 ). - Vidyadip

Question

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

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Answer

$f(x)=\cos ^{-1}\left(x^2-4\right)$ will be defined if $-1 \leq x^2-4 \leq 1$$
\begin{array}{ll}
\Rightarrow & -1+4 \leq x^2-4+4 \leq 1+4 \\
\Rightarrow & 3 \leq x^2 \leq 5 \\
\Rightarrow & x \in[-\sqrt{5},-\sqrt{3}] \cup[\sqrt{3}, \sqrt{5}] \\
& \left\{\because a^2 \leq x^2 \leq b^2 \Leftrightarrow x \in[-b,-a] \cup[a, b\right.
\end{array}
$

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