Range of the function `f (x) = cos^-1 (-{x})`, where `{.}` is fractional part function, is:

Options

A. ` ((pi)/(2), pi)`
B. ` ((pi)/(2), pi]`
C. ` [(pi)/(2), pi)`
D. `(0, (pi)/(2)] `

Answer

To find the range of the function \( f(x) = \cos^{-1}(-\{x\}) \), where \(\{x\}\) is the fractional part of \(x\), we will follow these steps:

### Step 1: Understand the fractional part function
The fractional part of \(x\), denoted as \(\{x\}\), is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
This means \(\{x\}\) takes values in the interval \([0, 1)\).

### Step 2: Analyze the expression \(-\{x\}\)
Since \(\{x\}\) ranges from \(0\) to just below \(1\), the expression \(-\{x\}\) will range from:
\[ -\{x\} \in (-1, 0] \]
This means \(-\{x\}\) takes values from just below \(0\) to \(-1\) (not including \(-1\)).

### Step 3: Determine the range of \(\cos^{-1}(-\{x\})\)
The function \(\cos^{-1}(y)\) is defined for \(y \in [-1, 1]\) and it is a decreasing function. The range of \(\cos^{-1}(y)\) is \([0, \pi]\).

Now we need to find the corresponding values of \(f(x) = \cos^{-1}(-\{x\})\) based on the range of \(-\{x\} \in (-1, 0]\).

1. When \(-\{x\} = 0\):
\[ f(x) = \cos^{-1}(0) = \frac{\pi}{2} \]
2. When \(-\{x\} \to -1\) (approaching -1 but not including):
\[ f(x) = \cos^{-1}(-1) = \pi \]

### Step 4: Combine the results
Thus, the range of \(f(x)\) is:
\[ \left[\frac{\pi}{2}, \pi\right) \]
This means \(f(x)\) can take values from \(\frac{\pi}{2}\) (inclusive) to \(\pi\) (exclusive).

### Final Answer
The range of the function \( f(x) = \cos^{-1}(-\{x\}) \) is:
\[ \left[\frac{\pi}{2}, \pi\right) \]