Question

6 Let $f(x)=\sqrt{\left(\sin ^{-1} x-\cos ^{-1} x\right)}, g(x)=\sqrt{\left(\tan ^{-1} x-\cot ^{-1} x\right)}$, then number of integer(s) in the domain of $f(x)+g(x)$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

$ \sin ^{-1} x-\cos ^{-1} x=2 \sin ^{-1} x-\frac{x}{2}, f(x)=\sqrt{\left(\sin ^{-1} x-\frac{\pi}{2}\right)}$
so $ \sin ^{-1} x \geq \frac{\pi}{4} \Rightarrow \frac{1}{\sqrt{2}} \leq x \leq 1 $ similarly
$g(x)=\sqrt{\left(2 \tan ^{-1} x-\frac{\pi}{2}\right)}, \tan ^{-1} x \geq \frac{\pi}{4} $ so $ 1 \leq x<\infty$
Domain of $( f + g ) x \Rightarrow\{1\} $