Question Number 180391 by Shrinava last updated on 11/Nov/22
$$\mathrm{y}\:=\:\mathrm{x}\:−\:\frac{\mathrm{2}}{\mathrm{x}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}. \\ $$
Commented by Shrinava last updated on 11/Nov/22
$$\mathrm{Answer}:\:\:\left(−\infty\:;\:+\infty\right) \\ $$
Answered by aleks041103 last updated on 12/Nov/22
$${x}\in\mathbb{R}\Rightarrow\forall{y}\in{I},\exists{x}\in\mathbb{R},{y}={x}−\frac{\mathrm{2}}{{x}} \\ $$$$\Rightarrow{x}^{\mathrm{2}} −{yx}−\mathrm{2}=\mathrm{0} \\ $$$${for}\:{this}\:{to}\:{have}\:{a}\:{soln}\:{in}\:{the}\:\mathbb{R}: \\ $$$${D}={y}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{1}\right)\left(−\mathrm{2}\right)={y}^{\mathrm{2}} +\mathrm{8}>\mathrm{0}\Leftrightarrow{y}\in\mathbb{R} \\ $$$$\Rightarrow{ans}.\:{I}=\left(−\infty,+\infty\right)\equiv\mathbb{R} \\ $$$$ \\ $$
Commented by Shrinava last updated on 12/Nov/22
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$