Find the domain of f(x)


Find the domain of $f(x)=2 \cos ^{-1} 2 x+\sin ^{-1} x$.


For $2 \cos ^{-1} 2 x$ to be defined.

$-1 \leq 2 x \leq 1$

$\Rightarrow-\frac{1}{2} \leq x \leq \frac{1}{2} \quad \ldots . .$ (i)

For $\sin ^{-1} x$ to be defined.

$-1 \leq x \leq 1 \quad \ldots$ (ii)

Domain of $f(x)=\left[-\frac{1}{2}, \frac{1}{2}\right] \cap[-1,1]$

$=\left[-\frac{1}{2}, \frac{1}{2}\right]$.

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