Find the domain of definition of the following functions :
`f(x)=sqrt(log_((1)/(2))(2x-3))`

Answer

To find the domain of the function \( f(x) = \sqrt{\log_{\frac{1}{2}}(2x - 3)} \), we need to ensure that the expression inside the square root is non-negative. This means we need to solve the inequality:

\[ \log_{\frac{1}{2}}(2x - 3) \geq 0 \]

### Step 1: Understand the logarithmic condition
The logarithm \( \log_{\frac{1}{2}}(y) \) is non-negative when \( y \) is in the interval \( (0, 1] \). Since the base \( \frac{1}{2} \) is less than 1, the logarithm is positive for values of \( y \) between 0 and 1.

### Step 2: Set up the inequalities
We need to find when \( 2x - 3 \) is in the interval \( (0, 1] \). This gives us two inequalities to solve:

1. \( 2x - 3 > 0 \)
2. \( 2x - 3 \leq 1 \)

### Step 3: Solve the first inequality
For the first inequality:

\[ 2x - 3 > 0 \]
Adding 3 to both sides:

\[ 2x > 3 \]
Dividing by 2:

\[ x > \frac{3}{2} \]

### Step 4: Solve the second inequality
For the second inequality:

\[ 2x - 3 \leq 1 \]
Adding 3 to both sides:

\[ 2x \leq 4 \]
Dividing by 2:

\[ x \leq 2 \]

### Step 5: Combine the results
Now we combine the results from both inequalities:

\[ \frac{3}{2} < x \leq 2 \]

### Step 6: Write the domain in interval notation
The domain of the function \( f(x) \) can be expressed in interval notation as:

\[ \left( \frac{3}{2}, 2 \right] \]

### Final Answer
Thus, the domain of the function \( f(x) = \sqrt{\log_{\frac{1}{2}}(2x - 3)} \) is:

\[ x \in \left( \frac{3}{2}, 2 \right] \]