Question

The domain of the function ${\text{cosec}}^{-1}\left(\frac{1+x}{x}\right)$ is:

• A. $[-\frac{1}{2},0)\cup [1,\infty )$
• B. $[-\frac{1}{2},\infty )-\left\{0\right\}$
• C. $(-1,-\frac{1}{2}]\cup (0,\infty )$
• D. $(-\frac{1}{2},\infty )-\left\{0\right\}$

Answer 1

The correct option is B $[-\frac{1}{2},\infty )-\left\{0\right\}$

For domain
$\frac{1+x}{x}\le -1\text{or}\frac{1+x}{x}\ge 1$
$\Rightarrow \frac{1+2x}{x}\le 0\text{or}\frac{1}{x}\ge 0$
$\Rightarrow x\in [-\frac{1}{2},0)\text{or}x\in (0,\infty )$
$\therefore$ Domain, $x\in [-\frac{1}{2},0)\cup (0,\infty )$
i.e. $x\in [-\frac{1}{2},\infty )-\left\{0\right\}$