Question

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

• A. Injective but not surjective
• B. Surjective but not injective
• C. Neither injective nor surjective
• D. Invertible

Answer 1

Answer: (B)

$f\left(x\right)=\frac{x}{1+x^2}$
$f^{\prime }\left(x\right)=\frac{{\left(1+x\right)}^2\cdot 1-x\cdot 2x}{{\left(1+x^2\right)}^2}=\frac{1-x^2}{{\left(1+x^2\right)}^2}$
f'(x) changes sign in different intervals.
$\therefore$ Not injective.
$y=\frac{x}{1+x^2}$
$y{x}^2-x+y=0$
For y $\ne$ 0
$D=1-4y^2\ge 0\Rightarrow y\in \left[-\frac{1}{2},\frac{1}{2}\right]-\left\{0\right\}$
For, y = 0 $\Rightarrow x=0$
$\therefore$ Part of range
$\therefore$ Range : $\left[-\frac{1}{2},\frac{1}{2}\right]$
$\therefore$ Surjective but not injective.