Find the domain of each of the following functions:
(i) $f(x)=\sin ^{-1} x^{2}$
(ii) $f(x)=\sin ^{-1} x+\sin x$
(iii) $f(x) \sin ^{-1} \sqrt{x^{2}-1}$
(iv) $f(x)=\sin ^{-1} x+\sin ^{-1} 2 x$
(i)
To the domain of $\sin ^{-1} y$ which is $[-1,1]$
$\therefore x^{2} \in[0,1]$ as $x^{2}$ can not be negative
$\therefore x \in[-1,1]$
Hence, the domain is $[-1,1]$
(ii)
Let $f(x)=g(x)+h(x)$, where
Therefore, the domain of $f(x)$ is given by the intersection of the domain of $g(x)$ and $h(x)$
The domain of $g(x)$ is $[-1,1]$
The domain of $h(x)$ is $(-\infty, \infty)$
Therfore, the intersection of $g(x)$ and $h(x)$ is $[-1,1]$
Hence, the domain is $[-1,1]$.
(iii)
To the domain of $\sin ^{-1} y$ which is $[-1,1]$
$\therefore x^{2}-1 \in[0,1]$ as square root can not be negative
$\Rightarrow x^{2} \in[1,2]$
$\Rightarrow x \in[-\sqrt{2},-1] \cup[1, \sqrt{2}]$
Hence, the domain is $[-\sqrt{2},-1] \cup[1, \sqrt{2}]$
(iv)
Let $f(x)=g(x)+h(x)$, where
Therefore, the domain of $f(x)$ is given by the intersection of the domain of $g(x)$ and $h(x)$
The domain of $g(x)$ is $[-1,1]$
The domain of $h(x)$ is $\left[-\frac{1}{2}, \frac{1}{2}\right]$
Therfore, the intersection of $g(x)$ and $h(x)$ is $\left[-\frac{1}{2}, \frac{1}{2}\right]$
Hence, the domain is $\left[-\frac{1}{2}, \frac{1}{2}\right]$
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