The domain of the function `f(x)=cos^(-1)[secx]`, where [x] denotes the greatest integer less than or equal to x, is

Options

A. `{x:x=(2n+1) pi, n in Z} cup { x : 2 m pi le x lt 2m pi+(pi)/(3), m in Z}`
B. `{x:x=2n pi , n in Z} cup { x: 2 m pi lt(x =2n pi) , n in Z} cup { x : 2 m pi lt x lt 2m pi +(pi)/(3), m in Z }`
C. `{x:(2n+1) pi, n in Z} cup { x : 2m pi lt x lt 2 m pi +(pi)/(3), m in Z}`
D. none of these

Answer

To find the domain of the function \( f(x) = \cos^{-1}[\sec x] \), where \([\cdot]\) denotes the greatest integer function, we need to determine the values of \( x \) for which the expression inside the inverse cosine function is valid.

### Step 1: Understand the Range of \(\sec x\)

The secant function, \(\sec x\), is defined as:

\[ \sec x = \frac{1}{\cos x} \]

The range of \(\sec x\) is:

\[ (-\infty, -1] \cup [1, \infty) \]

This means that \(\sec x\) can take values less than or equal to -1 or greater than or equal to 1.

### Step 2: Determine the Range of \([\sec x]\)

Since we are interested in \([\sec x]\), we need to find the integer values that \(\sec x\) can take.

1. For \(\sec x \leq -1\), the greatest integer function \([\sec x]\) can take values of \(-1, -2, -3, \ldots\).
2. For \(\sec x \geq 1\), the greatest integer function \([\sec x]\) can take values of \(1, 2, 3, \ldots\).

### Step 3: Determine Valid Inputs for \(\cos^{-1}\)

The function \(\cos^{-1}(y)\) is defined for \(y\) in the interval \([-1, 1]\). Therefore, we need to find the values of \([\sec x]\) that fall within this range:

\[ -1 \leq [\sec x] \leq 1 \]

From our previous analysis, we see that:

- The only valid integer from the range of \([\sec x]\) that satisfies this condition is \(-1\) and \(0\) (but \(0\) is not possible since \(\sec x\) cannot be between -1 and 1).

### Step 4: Finding Conditions for \([\sec x] = -1\)

For \([\sec x] = -1\):

\[ -1 \leq \sec x < 0 \]

However, since \(\sec x\) cannot be between -1 and 0, the only possibility is:

\[ \sec x = -1 \]

### Step 5: Solve for \(x\)

The equation \(\sec x = -1\) implies:

\[ \cos x = -1 \]

The solutions to \(\cos x = -1\) are:

\[ x = (2n + 1)\pi, \quad n \in \mathbb{Z} \]

### Step 6: Final Domain

Thus, the domain of the function \( f(x) = \cos^{-1}[\sec x] \) is:

\[ x = (2n + 1)\pi, \quad n \in \mathbb{Z} \]

### Summary of Steps

1. Determine the range of \(\sec x\).
2. Identify the possible values of \([\sec x]\).
3. Find the valid inputs for \(\cos^{-1}\).
4. Solve for \(x\) when \([\sec x] = -1\).
5. Conclude with the domain.