Question

If $f\left(x\right)=3{\sin }^{-1}x-2{\cos }^{-1}x$, then $f\left(x\right)$ is

• A. even function
• B. odd function
• C. neither even nor odd function
• D. both even and odd function

Answer 1

The correct option is C neither even nor odd function

Given : $f\left(x\right)=3{\sin }^{-1}x-2{\cos }^{-1}x$
Now, $f(-x)=3{\sin }^{-1}(-x)-2{\cos }^{-1}(-x)$
we know that,
${\sin }^{-1}(-x)=-{\sin }^{-1}x$ and
${\cos }^{-1}(-x)=\pi -{\cos }^{-1}x;-1\le x\le 1$
$\Rightarrow f(-x)=-3{\sin }^{-1}x-2(\pi -{\cos }^{-1}x)$
$\Rightarrow f(-x)=-3{\sin }^{-1}x+2{\cos }^{-1}x-2\pi$
$\because f(-x)\ne f\left(x\right)$ or $-f\left(x\right)$
$\therefore f\left(x\right)$ is neither even nor odd function.