Question

Find the range of the following functions
$\frac{1}{2-\cos 3x}$

Answer 1

$f\left(x\right)=\frac{1}{2-\cos 3x}$

$-1\le \cos 3x\le 1$

$⟹2-1\le 2-cos3x\le 2+1$

$⟹1\le 2-cos3x\le 3$

$⟹1\ge \frac{1}{2-\cos 3x}\ge \frac{1}{3}$

$⟹\frac{1}{3}\le \frac{1}{2-\cos 3x}\le 1$

$Range=[\frac{1}{3},1]$