To find the range of the function \( f(x) = \sin^{-1} x + |\sin^{-1} x| + \sin^{-1} |x| \), we will analyze the function for two cases: when \( x \geq 0 \) and when \( x < 0 \).
### Step 1: Case 1 - \( x \geq 0 \)
For \( x \geq 0 \):
- \( \sin^{-1} x \) is defined and non-negative since \( x \) is in the interval \([0, 1]\).
- Thus, \( |\sin^{-1} x| = \sin^{-1} x \).
- Also, \( \sin^{-1} |x| = \sin^{-1} x \) since \( |x| = x \).
Therefore, in this case:
\[
f(x) = \sin^{-1} x + \sin^{-1} x + \sin^{-1} x = 3 \sin^{-1} x
\]
### Step 2: Determine the range for \( x \in [0, 1] \)
The function \( \sin^{-1} x \) ranges from \( 0 \) to \( \frac{\pi}{2} \) as \( x \) goes from \( 0 \) to \( 1 \). Therefore:
- When \( x = 0 \), \( f(0) = 3 \cdot \sin^{-1}(0) = 0 \).
- When \( x = 1 \), \( f(1) = 3 \cdot \sin^{-1}(1) = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \).
Thus, for \( x \geq 0 \), the range of \( f(x) \) is:
\[
[0, \frac{3\pi}{2}]
\]
### Step 3: Case 2 - \( x < 0 \)
For \( x < 0 \):
- \( \sin^{-1} x \) is negative, so \( |\sin^{-1} x| = -\sin^{-1} x \).
- Since \( x \) is negative, \( |x| = -x \), thus \( \sin^{-1} |x| = \sin^{-1} (-x) = -\sin^{-1} x \).
Therefore, in this case:
\[
f(x) = \sin^{-1} x - \sin^{-1} x - \sin^{-1} x = -\sin^{-1} x
\]
### Step 4: Determine the range for \( x \in [-1, 0) \)
The function \( \sin^{-1} x \) ranges from \( -\frac{\pi}{2} \) to \( 0 \) as \( x \) goes from \( -1 \) to \( 0 \). Therefore:
- When \( x = -1 \), \( f(-1) = -\sin^{-1}(-1) = -(-\frac{\pi}{2}) = \frac{\pi}{2} \).
- When \( x \) approaches \( 0 \), \( f(x) \) approaches \( -\sin^{-1}(0) = 0 \).
Thus, for \( x < 0 \), the range of \( f(x) \) is:
\[
(0, \frac{\pi}{2}]
\]
### Step 5: Combine the ranges
Now we combine the ranges from both cases:
- From Case 1 (for \( x \geq 0 \)): \( [0, \frac{3\pi}{2}] \)
- From Case 2 (for \( x < 0 \)): \( (0, \frac{\pi}{2}] \)
The overall range of \( f(x) \) is:
\[
[0, \frac{3\pi}{2}]
\]
### Conclusion
Thus, the range of \( f(x) \) is \( [0, \frac{3\pi}{2}] \), which corresponds to option (b).
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