`f(x)=sin^-1x +|sin^-1x| +sin^-1|x| ` no. of solution of equation f(x)=x is (a) 1 (b) 0 (c) 2 (d) 3

Options

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

Explanation

To solve the equation \( f(x) = x \) where \( f(x) = \sin^{-1} x + |\sin^{-1} x| + \sin^{-1} |x| \), we will analyze the function \( f(x) \) in two cases based on the sign of \( x \).

### Step 1: Define the Function for \( x \geq 0 \)

For \( x \geq 0 \):
- \( \sin^{-1} x \) is non-negative.
- Therefore, \( |\sin^{-1} x| = \sin^{-1} x \).
- Also, \( \sin^{-1} |x| = \sin^{-1} x \) since \( |x| = x \).

Thus, for \( x \geq 0 \):
\[ f(x) = \sin^{-1} x + \sin^{-1} x + \sin^{-1} x = 3 \sin^{-1} x \]

### Step 2: Define the Function for \( x < 0 \)

For \( x < 0 \):
- \( \sin^{-1} x \) is negative.
- Therefore, \( |\sin^{-1} x| = -\sin^{-1} x \).
- Also, \( \sin^{-1} |x| = \sin^{-1} (-x) = -\sin^{-1} x \).

Thus, for \( x < 0 \):
\[ f(x) = \sin^{-1} x - \sin^{-1} x - \sin^{-1} x = -\sin^{-1} x \]

### Step 3: Analyze the Function

Now we have:
- For \( x \geq 0 \): \( f(x) = 3 \sin^{-1} x \)
- For \( x < 0 \): \( f(x) = -\sin^{-1} x \)

### Step 4: Find the Intersection Points with \( y = x \)

**Case 1: \( x \geq 0 \)**

We need to solve:
\[ 3 \sin^{-1} x = x \]
The function \( \sin^{-1} x \) is defined for \( x \in [0, 1] \) and increases from \( 0 \) to \( \frac{\pi}{2} \). Therefore, \( 3 \sin^{-1} x \) will increase from \( 0 \) to \( \frac{3\pi}{2} \).

- At \( x = 0 \), \( f(0) = 0 \).
- At \( x = 1 \), \( f(1) = \frac{3\pi}{2} \).

The line \( y = x \) intersects the curve \( y = 3 \sin^{-1} x \) at the origin (0,0) and will touch it at one point since \( 3 \sin^{-1} x \) grows faster than \( x \) for \( x > 0 \).

**Case 2: \( x < 0 \)**

We need to solve:
\[ -\sin^{-1} x = x \]
This equation can be rewritten as:
\[ \sin^{-1} x = -x \]
The function \( -x \) is a line that decreases from \( 0 \) to \( -1 \) as \( x \) goes from \( 0 \) to \( -1 \). The function \( \sin^{-1} x \) is defined for \( x \in [-1, 0] \) and increases from \( -\frac{\pi}{2} \) to \( 0 \).

- At \( x = -1 \), \( \sin^{-1}(-1) = -\frac{\pi}{2} \).
- At \( x = 0 \), \( \sin^{-1}(0) = 0 \).

The line \( y = -x \) intersects the curve \( y = \sin^{-1} x \) at one point in this interval.

### Conclusion

Thus, we have found:
- One solution at \( x = 0 \) for \( x \geq 0 \).
- One solution for \( x < 0 \).

Therefore, the total number of solutions to the equation \( f(x) = x \) is:
\[ \text{Total solutions} = 1 + 1 = 2 \]

The answer is **(c) 2**.