If f(x)=`sin^(-1)x` and g(x)=[sin(cosx)]+[cos(sinx)], then range of f(g(x)) is (where `[*]` denotes greatest integer function)

Options

A. `{(-pi)/2,pi/2}`
B. `{(-pi)/2,0}`
C. `{0,pi/2}`
D. `{-pi/2,0,pi/2}`

Answer

To find the range of the composite function \( f(g(x)) \), where \( f(x) = \sin^{-1}(x) \) and \( g(x) = [\sin(\cos x)] + [\cos(\sin x)] \), we will follow these steps:

### Step 1: Determine the range of \( g(x) \)

1. **Understanding \( g(x) \)**:
- The function \( g(x) \) is defined as the sum of two greatest integer functions: \( [\sin(\cos x)] \) and \( [\cos(\sin x)] \).
- We need to find the range of both \( \sin(\cos x) \) and \( \cos(\sin x) \).

2. **Finding the range of \( \sin(\cos x) \)**:
- The range of \( \cos x \) is \([-1, 1]\).
- Therefore, \( \sin(\cos x) \) will take values in the interval \([\sin(-1), \sin(1)]\).
- Calculating these values:
- \( \sin(-1) \approx -0.8415 \)
- \( \sin(1) \approx 0.8415 \)
- Thus, the range of \( \sin(\cos x) \) is approximately \([-0.8415, 0.8415]\).

3. **Finding the range of \( \cos(\sin x) \)**:
- The range of \( \sin x \) is also \([-1, 1]\).
- Therefore, \( \cos(\sin x) \) will take values in the interval \([\cos(-1), \cos(1)]\).
- Calculating these values:
- \( \cos(-1) \approx 0.5403 \)
- \( \cos(1) \approx 0.5403 \)
- Thus, the range of \( \cos(\sin x) \) is approximately \([0.5403, 1]\).

4. **Combining the ranges**:
- The possible values for \( g(x) = [\sin(\cos x)] + [\cos(\sin x)] \) can be:
- If both values are less than 0, \( g(x) = 0 \).
- If \( \sin(\cos x) \) is between 0 and 1, and \( \cos(\sin x) \) is between 0 and 1, \( g(x) \) can be 1 or 2.
- Therefore, the possible integer values for \( g(x) \) are \( 0, 1, \) and \( 2 \).

### Step 2: Determine the range of \( f(g(x)) \)

1. **Finding \( f(g(x)) \)**:
- Now we evaluate \( f(g(x)) = \sin^{-1}(g(x)) \).
- We need to find \( f(0), f(1), \) and \( f(2) \):
- \( f(0) = \sin^{-1}(0) = 0 \)
- \( f(1) = \sin^{-1}(1) = \frac{\pi}{2} \)
- \( f(2) \) is not defined since the domain of \( \sin^{-1}(x) \) is \([-1, 1]\).

2. **Conclusion**:
- The range of \( f(g(x)) \) is therefore \( \{0, \frac{\pi}{2}\} \).

### Final Answer:
The range of \( f(g(x)) \) is \( [0, \frac{\pi}{2}] \).

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