Question

Let $f(x)=\sqrt{\left({\sin }^{-1}x-{\cos }^{-1}x\right)},g(x)=\sqrt{\left({\tan }^{-1}x-{\cot }^{-1}x\right)}$ and $h(x)=$ $\sqrt{\left({\sec }^{-1}x-{\mathrm{cosec}}^{-1}x\right)}$ then correct statement(s) is(are)

• A. Domain of $f(x)+g(x)$ is $\{1\}$.
• B. Domain of $g(x)+h(x)$ is $[\sqrt{2},\infty )$
• C. Domain of $h(x)+f(x)$ if $\left[\frac{1}{\sqrt{2}},1\right]$
• D. Domain of $f(x)+g(x)+h(x)$ is $\varphi$

Answer 1

Answer: (A), (B), (D)

${\sin }^{-1}-{\cos }^{-1}x=2{\sin }^{-1}x-\pi /2$
$f(x):\sqrt{\left(2{\sin }^{-1}x-\pi /2\right)}$ so $2{\sin }^{-1}x-\pi /2\ge 0\Rightarrow {\sin }^{-1}x\ge \pi /4$
so Domain of, $f(x):\frac{1}{\sqrt{2}}\le x\le 1$ similarly $g(x)=\sqrt{\left(2{\tan }^{-1}x-\frac{\pi }{2}\right)},{\tan }^{-1}x\ge \frac{\pi }{4}$
so Domain of, $g(x):1\le x<\infty$ similarly $h(x)=\sqrt{\left(2{\sec }^{-1}x-\frac{\pi }{2}\right)},{\sec }^{-1}x\ge \frac{\pi }{4}$
so Domain of, $h(x):(-\infty ,-1]\cup [\sqrt{2},\infty )$ so Domain of $f(x)+g(x)$ is 1 only.
for $g(x)+h(x)$ Domain will be $[\sqrt{2},\infty )$,
for $h(x)+f(x)$, Domain will be null set
for $g(x)+h(x)+f(x)$, Domain will be null set .