Question

Let $f:R\to R$ be the function defined by $f\left(x\right)=\frac{1}{2-\cos x}\forall x\in R.$ Then, find the range of $f$.

Answer 1

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

Let $y=f\left(x\right)$.

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

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

or, $\cos x=2-\frac{1}{y}$.

We've,

$|\cos x|\le 1$, $\forall x\in R$

$-1\le 2-\frac{1}{y}\le 1$

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

$\frac{1}{3}\le y\le 1$.

So $y\in [\frac{1}{3},1]$.