If $f: R \rightarrow R$ and is defined by $f(x)=\frac{1}{2-\cos 3 x}$ for each $x \in R$, then the range of…

If $f: R \rightarrow R$ and is defined by $f(x)=\frac{1}{2-\cos 3 x}$ for each $x \in R$, then the range of $f$ is
  1. $(1 / 3,1)$
  2. $[1 / 3,1]$
  3. $(1,2)$
  4. $[1,2]$

Solution

$\begin{array}{lrl}\text { Given, } & f(x)= & \frac{1}{2-\cos 3 x} \\ \because & & -1 \leq \cos 3 x \leq 1 \\ \Rightarrow & 1 \leq-\cos 3 x \leq-1 \\ \Rightarrow & 2+1 & \leq 2-\cos 3 x \leq 2-1 \\ \Rightarrow & 3 \leq 2-\cos 3 x \leq 1 \\ \Rightarrow & \frac{1}{3} \leq \frac{1}{2-\cos 3 x} \leq 1 \\ \therefore & \text { Range of } f \text { is }\left[\frac{1}{3}, 1\right]\end{array}$

Asked in: AP EAMCET 2007

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