Question

Domain of \(cos^{-1} [x]\), where \([. ]\) denotes a greatest integer function

• A. \((-1, 2]\)
• B. \([-1,2]\)
• C. \((-1, 2)\)
• D. \([-1, 2)\)

Answer 1

The Correct Option is D

The greatest integer function [x] takes any real number x and rounds it down to the nearest integer.
The range of the greatest integer function is the set of all integers.
The domain of the inverse cosine function, \(\cos^{-1}(x)\), is [-1, 1], where x is a real number.
To find the domain of \(\cos^{-1}[x]\), we need to determine the values of x for which [x] lies within the range of [-1, 1].
Since the range of [x] is the set of all integers, we need to find the integers that lie within the range of [-1, 1].
The integers that lie within the range of [-1, 1] are -1, 0, and 1.
Therefore, the domain of \(\cos^{-1}[x]\) is the set of values for which [x] is equal to -1, 0, or 1.
So, the domain of \(\cos^{-1}[x]\) is [-1, 2) (option D).