Mathematics - Functions Question with Solution | TestHub

$f(x)=\frac{x}{1+x^2},x\in R$ Let, $y=\frac{x}{1+x^2}$ $\Rightarrow y{x}^2-x+y=0\Rightarrow x=\frac{1\pm \sqrt{1-4y^2}}{2}$ $\Rightarrow 1-4y^2\ge 0$ $\Rightarrow 1\ge 4y^2$ $\Rightarrow |y|\le \frac{1}{2}$ $\Rightarrow -\frac{1}{2}\le y\le \frac{1}{2}$ $\Rightarrow$ The range of $f$ is $\left[-\frac{1}{2},\frac{1}{2}\right]$.