Question

The function $f:R\to [-\frac{1}{2},\frac{1}{2}]$ defined as $f\left(x\right)=\frac{x}{1+x^2}$ is:

• A. invertible
• B. injective but not surjective
• C. surjective but not injective
• D. neither injective nor surjective

Answer 1

Answer: (C)

surjective but not injective

The domain of given function is R

When $x$ = - 1, then

$f($x$)$ = $\frac{-1}{2}$

When $x$= 1, then

$f($x$)$ = $\frac{1}{2}$

When $x$ $\in$ (-1,1)

$f($x$)$ $\in$ ($\frac{-1}{2}$, $\frac{1}{2}$)

When $x$ $\in$ (-$\infty$, - 1) $\cup$ (1,$\infty$), then

$f($x$)$ $\in$ ($\frac{-1}{2}$, $\frac{1}{2}$)

Therefore for all values of domain, the range of function $f($x$)$ lies between [$\frac{-1}{2}$, $\frac{1}{2}$]

=> $f($x$)$ $\in$ [$\frac{-1}{2}$, $\frac{1}{2}$]

Therefore, the given function is subjective and is not injective