If a function is defined as `f(x)=sqrt(log_(h(x))g(x))`, where `g(x)=|sinx|+sinx,h(x)=sinx+cosx,0lexlepi`. Then find th domain of `f(x)`.

Explanation

To find the domain of the function \( f(x) = \sqrt{\log_{h(x)} g(x)} \), we need to analyze the components \( g(x) \) and \( h(x) \) given by:

- \( g(x) = |\sin x| + \sin x \)
- \( h(x) = \sin x + \cos x \)

We need to ensure that the expression inside the square root is non-negative, and the logarithm is defined. This leads us to the following conditions:

1. **Condition for the logarithm**:
\[ \log_{h(x)} g(x) \geq 0 \]
This implies \( g(x) > 0 \) and \( h(x) > 0 \) and also \( g(x) \neq 1 \) when \( h(x) = 1 \).

2. **Condition for the square root**:
\[ \log_{h(x)} g(x) \geq 0 \implies g(x) \geq 1 \text{ if } h(x) > 1 \text{ or } g(x) \leq 1 \text{ if } h(x) < 1 \]

### Step 1: Analyze \( g(x) \)

The function \( g(x) = |\sin x| + \sin x \) can be simplified based on the interval \( [0, \pi] \):

- In the interval \( [0, \pi] \), \( \sin x \) is non-negative, thus \( |\sin x| = \sin x \).
- Therefore, \( g(x) = \sin x + \sin x = 2\sin x \).

### Step 2: Find the conditions for \( g(x) \)

For \( g(x) > 0 \):
\[ 2\sin x > 0 \implies \sin x > 0 \implies x \in (0, \pi) \]

For \( g(x) = 1 \):
\[ 2\sin x = 1 \implies \sin x = \frac{1}{2} \implies x = \frac{\pi}{6} \]

### Step 3: Analyze \( h(x) \)

The function \( h(x) = \sin x + \cos x \) can be analyzed as follows:

- The maximum value occurs when \( x = \frac{\pi}{4} \) where \( h\left(\frac{\pi}{4}\right) = \sqrt{2} \).
- The minimum value occurs at the endpoints \( h(0) = 1 \) and \( h(\pi) = -1 \).

### Step 4: Find the conditions for \( h(x) \)

For \( h(x) > 0 \):
\[ \sin x + \cos x > 0 \]
This is satisfied in the intervals where both sine and cosine are positive or one is positive and the other is less than the absolute value of the positive one.

### Step 5: Solve for \( h(x) = 1 \)

Setting \( h(x) = 1 \):
\[ \sin x + \cos x = 1 \implies \sqrt{2}\sin\left(x + \frac{\pi}{4}\right) = 1 \]
This gives us \( x = \frac{\pi}{4} \).

### Step 6: Combine the conditions

- From \( g(x) > 0 \): \( x \in (0, \pi) \)
- From \( h(x) > 0 \): \( x \in (0, \frac{\pi}{2}) \)
- Exclude \( x = \frac{\pi}{6} \) where \( g(x) = 1 \) and \( x = \frac{\pi}{4} \) where \( h(x) = 1 \).

### Final Domain

Thus, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = (0, \frac{\pi}{6}) \cup (\frac{\pi}{6}, \frac{\pi}{4}) \cup (\frac{\pi}{4}, \frac{\pi}{2}) \]