Question

If $f : R \rightarrow\left[\frac{-\pi}{4}, \frac{\pi}{2}\right)$ defined as $f ( x )=\tan ^{-1}\left( x ^4- x ^2-\frac{7}{4}+\tan ^{-1} \alpha\right)$ is surjective function then $\alpha$ is equal to

• A. 1
• B. $\tan 1$
• C. $1+\tan 1$
• D. $1-\tan 1$

Answer 1

Answer: (B)

$\Theta f ( x )$ is surjective
$\therefore \text { Range of } f ( x )=\left[\frac{-\pi}{4}, \frac{\pi}{2}\right) $
$f ( x )=\tan ^{-1}\left(\left( x ^2-\frac{1}{2}\right)^2-2+\tan ^{-1} \alpha\right) $
$\therefore -2+\tan ^{-1} \alpha=-1 $
$\Rightarrow \tan ^{-1} \alpha=1 \Rightarrow \alpha=\tan 1$