Question

If $f:(-1,1)\to (-3,3)$ defined by $f\left(x\right)=\cos \left({\cos }^{-1}x\right)+\sin \left({\sin }^{-1}x\right)-\tan \left({\tan }^{-1}x\right)$, then $f$ is

• A. One-one, onto
• B. Many-one, onto
• C. Many-one, into
• D. One-one, into

Answer 1

The correct option is D One-one, into

$f\left(x\right)=\cos \left({\cos }^{-1}x\right)+\sin \left({\sin }^{-1}x\right)-\tan \left({\tan }^{-1}x\right)$
$\Rightarrow f\left(x\right)=x+x-x=x$
Clearly f is one-one
But Range $=(-1,1)\ne$ co-domain.
$\therefore f$ is one-one, into