what is the range of y = csc⁻¹(x)?\n-\\frac{\\pi}{2},\\frac{\\pi}{2}\n-\\frac{\\pi}{2},0) and…

what is the range of y = csc⁻¹(x)?\n-\\frac{\\pi}{2},\\frac{\\pi}{2}\n-\\frac{\\pi}{2},0) and (0,\\frac{\\pi}{2}\n0,\\pi\n0,\\frac{\\pi}{2}) and (\\frac{\\pi}{2},\\pi

what is the range of y = csc⁻¹(x)?\n-\\frac{\\pi}{2},\\frac{\\pi}{2}\n-\\frac{\\pi}{2},0) and (0,\\frac{\\pi}{2}\n0,\\pi\n0,\\frac{\\pi}{2}) and (\\frac{\\pi}{2},\\pi

Answer

Explanation:

Step1: Recall inverse - cosecant function properties

The inverse cosecant function (y = \csc^{-1}(x)) is the inverse of the cosecant function (y=\csc(x)) restricted to certain intervals. The domain of (y = \csc(x)) for the inverse - function definition is (\left[-\frac{\pi}{2},0\right)\cup\left(0,\frac{\pi}{2}\right]) and its range is ((-\infty,- 1]\cup[1,\infty)). The range of (y=\csc^{-1}(x)) is the domain of the restricted (y = \csc(x)) function.

Answer:

(\left[-\frac{\pi}{2},0\right)\text{ and }\left(0,\frac{\pi}{2}\right])