Question

The domain and range of the function $f\left(x\right)=\frac{1}{2-\cos 3x}$ is

• A. $R-\left\{2\right\},[\frac{1}{3},1]$
• B. $R,[\frac{1}{3},1]$
• C. $R,[\frac{-1}{3},\frac{1}{3}]$
  
• D. $R-\left\{2\right\},[\frac{-1}{3},\frac{1}{3}]$

Answer 1

The correct option is B $R,[\frac{1}{3},1]$

As $\cos 3x\ne 2$ for any $x\in R$
$f$ is defined for all $x\in R$
$\therefore D_f=R$

$-1\le -\cos 3x\le 1$
$\Rightarrow 1\le 2-\cos 3x\le 3$
$\Rightarrow \frac{1}{3}\le \frac{1}{2-\cos 3x}\le 1$
$\Rightarrow f\left(x\right)\in [\frac{1}{3},1]\forall x\in D_f$
$\therefore R_f=[\frac{1}{3},1]$