Question

Domain of $\cos^{-1}[x]$ is, where $[.]$ denotes greatest integer function:

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

Hint:

The greatest integer function $[x] = n$ implies $n \le x <n+1$. Always remember that the upper bound in the domain of $[x]$ results in an open bracket ($2$ is not included because $[2]=2$, which is outside the range $[-1, 1]$).

Answer 1

The Correct Option is D

To determine the domain of the function \(\cos^{-1}[x]\), we must consider both the properties of the inverse cosine function and the greatest integer function.

The inverse cosine function, denoted as \(\cos^{-1}(y)\), is defined for \(y\) in the interval \([-1, 1]\) inclusive. This means that the function can only accept input values within this range. Therefore, for \(\cos^{-1}[x]\) to be defined, the expression inside the cosine inverse, which is \([x]\) (the greatest integer function of \(x\)), must also lie within this interval.

The greatest integer function \([x]\), also known as the floor function, gives the greatest integer less than or equal to \(x\). Therefore, if

• \([-1 \leq [x] \leq 1]\), then the domain of \(x\) must be such that when the floor function is applied, the result is \(-1, 0,\) or \(1\).

Now, let's examine when \([x]\) can take these values:

• For \([x] = -1\), \(x\) falls in the interval \([-1, 0)\).
• For \([x] = 0\), \(x\) falls in the interval \([0, 1)\).
• For \([x] = 1\), \(x\) falls in the interval \([1, 2)\).

Combining these intervals, the domain of \(\cos^{-1}[x]\) is \([-1, 2)\).

Thus, the correct answer is:

\([-1, 2)\)

Let's eliminate the incorrect options:

• \((-1, 2]\) includes numbers where the greatest integer is outside the domain of the \(\cos^{-1}(y)\) function.
• \([-1, 2]\) includes both endpoints, making it incorrect because \([x]\) cannot equal 2.
• \((-1, 2)\) excludes \(-1\), which is required for \([x] = -1\).