Let f be a real - valued function defined on R ( the set of real numbers) such that `f(x) = sin^(-1) ( sin x) + cos^(-1) ( cos x)`
Number of values of x in interval (0, 3) so that f(x) is an integer, is equal to

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the problem, we need to analyze the function \( f(x) = \sin^{-1}(\sin x) + \cos^{-1}(\cos x) \) and determine how many integer values it takes in the interval \( (0, 3) \).

### Step 1: Understanding the Function
The function consists of two parts:
1. \( \sin^{-1}(\sin x) \)
2. \( \cos^{-1}(\cos x) \)

### Step 2: Analyze \( \sin^{-1}(\sin x) \)
The function \( \sin^{-1}(\sin x) \) is defined as:
- \( x \) when \( x \) is in the interval \( [0, \frac{\pi}{2}] \)
- \( \pi - x \) when \( x \) is in the interval \( (\frac{\pi}{2}, \frac{3\pi}{2}] \)
- \( 2\pi - x \) when \( x \) is in the interval \( (\frac{3\pi}{2}, 2\pi] \)

### Step 3: Analyze \( \cos^{-1}(\cos x) \)
The function \( \cos^{-1}(\cos x) \) is defined as:
- \( x \) when \( x \) is in the interval \( [0, \pi] \)
- \( 2\pi - x \) when \( x \) is in the interval \( (\pi, 2\pi] \)

### Step 4: Define \( f(x) \) in the Interval \( (0, 3) \)
Now we need to evaluate \( f(x) \) in the interval \( (0, 3) \):
- For \( x \in (0, \frac{\pi}{2}) \), \( f(x) = x + x = 2x \)
- For \( x \in (\frac{\pi}{2}, 3) \):
- In \( (\frac{\pi}{2}, \pi) \), \( f(x) = (\pi - x) + x = \pi \)
- In \( (\pi, 3) \), \( f(x) = (2\pi - x) + x = 2\pi \)

### Step 5: Determine Integer Values
Now, we check for integer values of \( f(x) \):
1. **For \( x \in (0, \frac{\pi}{2}) \)**:
- \( f(x) = 2x \)
- \( 2x \) can take integer values when \( x = \frac{1}{2}, 1, \frac{3}{2} \) (as \( 2x \) must be in the range \( (0, \pi) \)).
- The integer values of \( 2x \) in this interval are \( 1 \) and \( 2 \).

2. **For \( x \in (\frac{\pi}{2}, \pi) \)**:
- \( f(x) = \pi \), which is not an integer.

3. **For \( x \in (\pi, 3) \)**:
- \( f(x) = 2\pi \), which is also not an integer.

### Step 6: Count the Integer Values
From the analysis:
- The integer values of \( f(x) \) occur only in the interval \( (0, \frac{\pi}{2}) \) and are \( 1 \) and \( 2 \).
- Therefore, the number of values of \( x \) in the interval \( (0, 3) \) such that \( f(x) \) is an integer is **3** (for \( x = \frac{1}{2}, 1, \frac{3}{2} \)).

### Final Answer
The number of values of \( x \) in the interval \( (0, 3) \) such that \( f(x) \) is an integer is **3**.

---