Question

Let $f\left(x\right)={\sin }^{-1}|\sin x|+{\cos }^{-1}\left(\cos x\right),x\in R.$ Which of the following statements is/are TRUE?

• A. $f\left(f\right(3\left)\right)=\pi$
• B. $f\left(x\right)$ is periodic with fundamental period $2\pi$.
• C. $f\left(x\right)$ is neither even nor odd.
• D. Range of $f\left(x\right)$ is $[0,2\pi ]$.

Answer 1

Answer: (A), (B)

$f\left(x\right)$ is periodic with fundamental period $2\pi$.

$f\left(x\right)={\sin }^{-1}|\sin x|+{\cos }^{-1}\left(\cos x\right)$
Clearly, $f\left(x\right)$ is periodic function with fundamental period $2\pi$.
So, we calculate $f$ in $[0,2\pi ]$
$f\left(x\right)=\{\begin{array}{cc}2x,& 0\le x\le \frac{\pi }{2}\\ \pi ,& \frac{\pi }{2}<x\le \frac{3\pi }{2}\\ 4\pi -2x,& \frac{3\pi }{2}<x\le 2\pi \end{array}$

The graph of $f\left(x\right)$ is shown below.


From the graph, range of $f\left(x\right)$ is $[0,\pi ]$

$f\left(3\right)=\pi$
$\Rightarrow f\left(f\right(3\left)\right)=f\left(\pi \right)=\pi$

$f(-x)=f\left(x\right),$ so $f\left(x\right)$ is even.