The natural domain of the function `f(x)=sqrt(sin^(-1)(2x)+(pi)/(3))` is

Options

A. `[-(1)/(2),(1)/(2)]`
B. `[-(sqrt3)/(4),(1)/(2)]`
C. `[(13)/(4),(1)/(2)]`
D. `[-(sqrt3)/(2),1]`

Answer

To find the natural domain of the function \( f(x) = \sqrt{\sin^{-1}(2x) + \frac{\pi}{3}} \), we need to ensure that the expression inside the square root is non-negative. Let's go through the steps to determine the domain.

### Step 1: Set up the inequality
We need the expression inside the square root to be greater than or equal to zero:
\[ \sin^{-1}(2x) + \frac{\pi}{3} \geq 0 \]

### Step 2: Isolate the sine inverse function
Subtract \(\frac{\pi}{3}\) from both sides:
\[ \sin^{-1}(2x) \geq -\frac{\pi}{3} \]

### Step 3: Understand the range of the sine inverse function
The function \(\sin^{-1}(y)\) is defined for \(y\) in the range \([-1, 1]\) and its output lies in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Therefore, we need to ensure that \(2x\) falls within the valid input range for the sine inverse function:
\[ -1 \leq 2x \leq 1 \]

### Step 4: Solve the inequalities
1. From \(2x \geq -1\):
\[ x \geq -\frac{1}{2} \]
2. From \(2x \leq 1\):
\[ x \leq \frac{1}{2} \]

### Step 5: Combine the results
Now we combine the results from the inequalities:
\[ -\frac{1}{2} \leq x \leq \frac{1}{2} \]

### Step 6: Verify the sine inverse condition
Next, we need to ensure that the condition \(\sin^{-1}(2x) \geq -\frac{\pi}{3}\) holds true within the range we found. The sine function is increasing, and we know:
\[ \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]
So we need:
\[ 2x \geq -\frac{\sqrt{3}}{2} \]
This simplifies to:
\[ x \geq -\frac{\sqrt{3}}{4} \]

### Step 7: Final domain
Now we have two conditions:
1. \(x \geq -\frac{1}{2}\)
2. \(x \geq -\frac{\sqrt{3}}{4}\)

The more restrictive condition is \(x \geq -\frac{\sqrt{3}}{4}\). Hence, we also need to consider the upper limit \(x \leq \frac{1}{2}\).

Thus, the final domain of the function is:
\[ \left[-\frac{\sqrt{3}}{4}, \frac{1}{2}\right] \]

### Final Answer
The natural domain of the function \( f(x) = \sqrt{\sin^{-1}(2x) + \frac{\pi}{3}} \) is:
\[ \left[-\frac{\sqrt{3}}{4}, \frac{1}{2}\right] \]

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