Question

Let $f(x)=\sin ^{-1}(\sin x), g(x)=\cos ^{-1}(\cos x)$, then which one of the following incorrect?

• A. $f(x)=g(x) \forall x \in\left(0, \frac{\pi}{4}\right)$
• B. $f ( x )< g ( x ) \forall x \in\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right)$
• C. $f ( x )< g ( x ) \forall x \in\left(\pi, \frac{5 \pi}{4}\right)$
• D. $f(x)>g(x) \forall x \in\left(\frac{3 \pi}{2}, 2 \pi\right)$

Answer 1

Answer: (D)

$f(x)=\sin ^{-1}(\sin x)=\begin{cases}x, & \frac{-\pi}{2} \leq x \leq \frac{\pi}{2} \\ \pi-x, & \frac{\pi}{2}< x \leq \frac{3 \pi}{2} \\ x-2 \pi, & \frac{3 \pi}{2}< x \leq \frac{5 \pi}{2}\end{cases}$
$g ( x )=\cos ^{-1}(\cos x )=\begin{cases} x , & 0 \leq x \leq \pi \\ 2 \pi- x , & \pi< x \leq 2 \pi\end{cases}$
Now, verify alternatives.