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).

Answer 2

Let \( f(x) = \cos^{-1}([x]) \), where \([x]\) is the greatest integer function. The domain of \( \cos^{-1}(x) \) is \( [-1, 1] \). So, \([x]\) must lie in \( [-1, 1] \). That means possible values of \([x]\) are \( -1, 0, 1 \) For:

• \([x] = -1 \Rightarrow x \in [-1, 0)\)
• \([x] = 0 \Rightarrow x \in [0,1)\)
• \([x] = 1 \Rightarrow x \in [1,2) 

Combining all: \[ x \in [-1, 2) \] So, the domain is: \([-1, 2)\)

Answer 3

We need to find the domain of \(cos^{-1}[x]\), where [x] denotes the greatest integer function.

The domain of the inverse cosine function, cos^-1(y), is -1 ≤ y ≤ 1. Therefore, for cos^-1[x] to be defined, we must have:

-1 ≤ [x] ≤ 1

This means that the greatest integer function of x must be between -1 and 1, inclusive. Let's consider the possible values of [x]:

• [x] = -1 => -1 ≤ x < 0
• [x] = 0 => 0 ≤ x < 1
• [x] = 1 => 1 ≤ x < 2

Combining these intervals, we get:

-1 ≤ x < 0, 0 ≤ x < 1, 1 ≤ x < 2

This is equivalent to -1 ≤ x < 2

Therefore, the domain of cos^-1[x] is [-1, 2).

Answer: [-1, 2)