Question

Find the domain and range of the following function:
$\frac{x}{2+x^2}$

Answer 1

Domain $x\in (-\infty ,\infty )$

Range $\Rightarrow y=\frac{x}{x^2+2}$

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

$n>0$

$1-4\left(2y\right)\left(y\right)\ge 0$

$1-8y^2\ge 0$

$8y^2-1\le 0$

$y^2-1/8\le 0$

$(y+1/2\sqrt{2})(y-1/2\sqrt{2})\le 0$

$y\in [\frac{-1}{2\sqrt{2}},\frac{1}{2\sqrt{2}}]$