Question

$f:(0,\infty )\to (-\frac{\pi }{2},\frac{\pi }{2})$ be defined as, $f\left(x\right)=arc\tan \left(x\right)$

The above function can be classified as

• A. injective but not surjective
• B. surjective but not injective
• C. neither injective nor surjective
• D. both injective as well as surjective

Answer 1

Answer: (D)

both injective as well as surjective

$f\left(x\right)={\tan }^{-1}x$
$f^{\prime }\left(x\right)=\frac{1}{1+x^2}>0\forall x\in R$
Thus $f\left(x\right)$ is injective.
Also $f\left(x\right)$ is surjective as co-domain and range are same.