The domain of the function

Question:

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

  1. $\left(-1,-\frac{1}{2}\right] \cup(0, \infty)$

  2. $\left[-\frac{1}{2}, 0\right) \cup[1, \infty)$

  3. $\left(-\frac{1}{2}, \infty\right)-\{0\}$

  4. $\left[-\frac{1}{2}, \infty\right)-\{0\}$


Correct Option: , 4

Solution:

$\frac{1+x}{x} \in(-\infty,-1] \cup[1, \infty)$

$\frac{1}{x} \in(-\infty,-2] \cup[0, \infty)$

$x \in\left[-\frac{1}{2}, 0\right) \cup(0, \infty)$

$x \in\left[-\frac{1}{2}, \infty\right)-\{0\}$

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