The domain of the function operatorname{cosec | vedclass

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(D) The domain of $\operatorname{cosec}^{-1}(y)$ is $y \in (-\infty, -1] \cup [1, \infty)$.For the function $\operatorname{cosec}^{-1}\left(\frac{1+x}

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

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(D) The domain of $\operatorname{cosec}^{-1}(y)$ is $y \in (-\infty, -1] \cup [1, \infty)$.For the function $\operatorname{cosec}^{-1}\left(\frac{1+x}{x}\right)$,we must have $\frac{1+x}{x} \in (-\infty, -1] \cup [1, \infty)$.Case $1$: $\frac{1+x}{x} \geq 1$$\frac{1}{x} + 1 \geq 1 \implies \frac{1}{x} \geq 0 \implies x > 0$.Case $2$: $\frac{1+x}{x} \leq -1$$\frac{1}{x} + 1 \leq -1 \implies \frac{1}{x} \leq -2$.Since $\frac{1}{x} \leq -2$,$x$ must be negative. Multiplying by $x$ (which is negative) reverses the inequality:$1 \geq -2x \implies x \geq -\frac{1}{2}$.Thus,$x \in [-\frac{1}{2}, 0)$.Combining both cases,the domain is $x \in [-\frac{1}{2}, 0) \cup (0, \infty)$,which can be written as $[-\frac{1}{2}, \infty) - \{0\}$.

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

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