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. inhective but not surjective
• C. surjective but not injective
• D. neither injective nor surjective.

Answer 1

The correct option is A invertible

given,

$f:R\to [-\frac{1}{2},\frac{1}{2}]$

$f\left(x\right)=\frac{x}{1+x^2}$

$\text{Let}f\left(x\right)=y$

$y+y{x}^2=x$

$y{x}^2-x+y=0$

$x\in \left[\frac{1\pm \sqrt{1-4y^2}}{2y}\right]$ forall $y\in [\frac{-1}{2},\frac{1}{2}]$

$\therefore f\left(x\right)$ is invertible in the range $[\frac{-1}{2},\frac{1}{2}]$

option $A$ is invertible