Question

Which of the following is/are correct for the function $f\left(x\right)=\frac{1}{3+2\cos x}$

• A. Domain is $R$
• B. Domain is $R-\{-\frac{3}{2}\}$
• C. Minimum value is $\frac{1}{5}$
• D. Maximum value is $1$

Answer 1

Answer: (A), (C), (D)

The correct options are
A Domain is $R$
C Minimum value is $\frac{1}{5}$
D Maximum value is $1$

$f\left(x\right)=\frac{1}{3+2\cos x}$
Clearly, $f\left(x\right)$ is defined for all real $x\in R$ as $\cos x\ne -\frac{3}{2}$
$\therefore$ Domain is $R$

We know that
$-1\le \cos x\le 1$
$\Rightarrow -2\le 2\cos x\le 2$
$\Rightarrow 1\le 3+2\cos x\le 5$
$\Rightarrow \frac{1}{5}\le \frac{1}{3+2\cos x}\le 1$
$\therefore$ Range $=[\frac{1}{5},1]$