Domain (D) and range (R ) of `f(x)=sin^(-1)(cos^(-1)[x])` where [ ] denotes the greatest integer function is

Options

A. `D-=[1, 2), R-={0}`
B. `D-=[0,1), R-={-1, 0,1}`
C. `D-=[-1, 1), R-={0, (pi)/(2), pi}`
D. `D-=[-1,1], R-={-(pi)/(2), 0, (pi)/(2)}`

Answer

To find the domain (D) and range (R) of the function \( f(x) = \sin^{-1}(\cos^{-1}[\lfloor x \rfloor]) \), where \( [\cdot] \) denotes the greatest integer function, we will analyze the components of the function step by step.

### Step 1: Determine the domain of the greatest integer function \( \lfloor x \rfloor \)

The greatest integer function \( \lfloor x \rfloor \) can take any integer value depending on the value of \( x \). Therefore, the domain of \( \lfloor x \rfloor \) is all real numbers:

**Domain of \( \lfloor x \rfloor \):** \( (-\infty, \infty) \)

### Step 2: Determine the range of \( \lfloor x \rfloor \)

The output of \( \lfloor x \rfloor \) is any integer \( n \) such that \( n \in \mathbb{Z} \). Thus, the range of \( \lfloor x \rfloor \) is:

**Range of \( \lfloor x \rfloor \):** \( \mathbb{Z} \)

### Step 3: Determine the domain of \( \cos^{-1}(y) \)

The function \( \cos^{-1}(y) \) is defined for \( y \) in the interval \( [-1, 1] \). Since \( \lfloor x \rfloor \) can take any integer value, we need to restrict \( \lfloor x \rfloor \) to the values within this interval. The only integers that fall within this range are \( 0 \) and \( 1 \) (as \( -1 \) is not included in the output of \( \lfloor x \rfloor \) for non-negative \( x \)).

Thus, the valid inputs for \( \cos^{-1}([\lfloor x \rfloor]) \) are:

- If \( \lfloor x \rfloor = 0 \) then \( \cos^{-1}(0) = \frac{\pi}{2} \)
- If \( \lfloor x \rfloor = 1 \) then \( \cos^{-1}(1) = 0 \)

### Step 4: Determine the range of \( \cos^{-1}([\lfloor x \rfloor]) \)

The range of \( \cos^{-1}([\lfloor x \rfloor]) \) is therefore:

**Range of \( \cos^{-1}([\lfloor x \rfloor]) \):** \( \{0, \frac{\pi}{2}\} \)

### Step 5: Determine the domain of \( \sin^{-1}(z) \)

The function \( \sin^{-1}(z) \) is defined for \( z \) in the interval \( [-1, 1] \). Since the outputs from \( \cos^{-1}([\lfloor x \rfloor]) \) are \( 0 \) and \( \frac{\pi}{2} \), we need to check if these values are valid inputs for \( \sin^{-1}(z) \):

- \( \sin^{-1}(0) = 0 \)
- \( \sin^{-1}(\frac{\pi}{2}) \) is not defined since \( \frac{\pi}{2} > 1 \)

Thus, the only valid input for \( \sin^{-1}(z) \) is \( 0 \).

### Step 6: Determine the range of \( f(x) \)

Since the only valid output from \( \sin^{-1}(z) \) is \( 0 \), the range of \( f(x) \) is:

**Range of \( f(x) \):** \( \{0\} \)

### Final Answer

**Domain (D) of \( f(x) \):** \( (-\infty, \infty) \)

**Range (R) of \( f(x) \):** \( \{0\} \)