Relations and Functions — Class 12 Maths Solution

Given function $f(x) = \frac{1}{{2 - \cos x}},\forall x \in R$

Let $y = \frac{1}{{2 - \cos x}}$

$\Rightarrow$ $2y - y\cos x = 1$

$\Rightarrow$ $y\cos x = 2y - 1$

$\Rightarrow$ $\cos x = \frac{{2y - 1}}{y} = 2 - \frac{1}{y} \Rightarrow \cos x = 2 - \frac{1}{y}$

$\Rightarrow$ $- 1 \le \cos x \le 1 \Rightarrow - 1 \le 2 - \frac{1}{y} \le 1$

$\Rightarrow$ $- 3 \le - \frac{1}{y} \le - 1 \Rightarrow 1 \le \frac{1}{y} \le 3$

$\Rightarrow$ $\frac{1}{3} \le \frac{1}{y} \le 1$
So, $$y$$ range is $\left[ {\frac{1}{3},1} \right]$.