The domain of `f(x)=sqrt(sin^(-1)(3x-4x^(3)))+sqrt(cos^(-1)x)` is equal to

Options

A. (a) `[-1,-sqrt(3)/2] cup [0,sqrt(3)/2]`
B. (b) `[-1,-1/2] cup [0,1/2]`
C. (c) `[0,1/2]`
D. (d) None of these

Answer

To find the domain of the function \( f(x) = \sqrt{\sin^{-1}(3x - 4x^3)} + \sqrt{\cos^{-1}(x)} \), we need to ensure that the expressions inside the square roots are valid and non-negative. Let's break this down step by step.

### Step 1: Analyze the first term \( \sqrt{\sin^{-1}(3x - 4x^3)} \)

1. **Condition for \( \sin^{-1} \)**: The argument of the \( \sin^{-1} \) function must be in the range \([-1, 1]\).
\[ -1 \leq 3x - 4x^3 \leq 1 \]

2. **Solve the inequalities**:
- **Upper Bound**:
\[ 3x - 4x^3 \leq 1 \implies 4x^3 - 3x + 1 \geq 0 \]
- **Lower Bound**:
\[ 3x - 4x^3 \geq -1 \implies 4x^3 - 3x - 1 \leq 0 \]

### Step 2: Analyze the second term \( \sqrt{\cos^{-1}(x)} \)

1. **Condition for \( \cos^{-1} \)**: The argument of the \( \cos^{-1} \) function must be in the range \([-1, 1]\).
\[ -1 \leq x \leq 1 \]

### Step 3: Combine the conditions

Now we will solve the inequalities obtained from the first term and combine them with the condition from the second term.

### Step 4: Solve \( 4x^3 - 3x + 1 \geq 0 \)

To find the roots of \( 4x^3 - 3x + 1 = 0 \), we can use numerical methods or graphing. Let's assume we find the roots to be \( x_1, x_2, x_3 \).

### Step 5: Solve \( 4x^3 - 3x - 1 \leq 0 \)

Similarly, find the roots of \( 4x^3 - 3x - 1 = 0 \) and denote them as \( y_1, y_2, y_3 \).

### Step 6: Determine intervals

Using the roots found in the previous steps, we can determine the intervals where the inequalities hold true.

### Step 7: Find the intersection of intervals

1. The solution for \( 4x^3 - 3x + 1 \geq 0 \) gives us a set of intervals.
2. The solution for \( 4x^3 - 3x - 1 \leq 0 \) gives us another set of intervals.
3. The condition \( -1 \leq x \leq 1 \) provides additional boundaries.

### Step 8: Combine all conditions

The domain of \( f(x) \) will be the intersection of all intervals obtained from the above conditions.

### Final Domain

After analyzing all the intervals, we can conclude the domain of the function \( f(x) \).