Question

The range of $f\left(x\right)=x+\frac{1}{x}\forall x\ne 0$ is

• A. $\left(-2,2\right)$
• B. $(-\infty ,-1]\cup [1,\infty )$
• C. $(-\infty ,-2]\cup [2,\infty )$

Answer 1

The correct option is C $(-\infty ,-2]\cup [2,\infty )$

Given: $f\left(x\right)=x+\frac{1}{x}\forall x\ne 0$
Let $t=x+\frac{1}{x}$
$\Rightarrow tx=x^2+1\Rightarrow x^2-tx+1=0$

$\because t$ is real.
$\Rightarrow$ The roots of this equation should exist.
or it's discriminant $D\ge 0$
$\Rightarrow t^2-4\ge 0\Rightarrow (t-2)(t+2)\ge 0\Rightarrow t\in (-\infty ,-2]\cup [2,\infty )$