Find the values of x which the function `f(x)=sqrt(log_(1//2)((x-1)/(x+5))` is defined.

Explanation

To find the values of \( x \) for which the function

\[ f(x) = \sqrt{\log_{\frac{1}{2}}\left(\frac{x-1}{x+5}\right)} \]

is defined, we need to ensure that the expression inside the square root is non-negative. This means we need to satisfy the following conditions:

1. The argument of the logarithm must be positive:
\[ \frac{x-1}{x+5} > 0 \]

2. The logarithm itself must be non-negative:
\[ \log_{\frac{1}{2}}\left(\frac{x-1}{x+5}\right) \geq 0 \]

### Step 1: Solve the inequality \( \frac{x-1}{x+5} > 0 \)

To determine when the fraction is positive, we analyze the signs of the numerator and denominator:

- The numerator \( x - 1 > 0 \) gives \( x > 1 \).
- The denominator \( x + 5 > 0 \) gives \( x > -5 \).

The fraction \( \frac{x-1}{x+5} \) is positive when both the numerator and denominator are either both positive or both negative.

**Case 1: Both positive**
- \( x - 1 > 0 \) implies \( x > 1 \)
- \( x + 5 > 0 \) is always true for \( x > 1 \)

Thus, for this case, \( x > 1 \).

**Case 2: Both negative**
- \( x - 1 < 0 \) implies \( x < 1 \)
- \( x + 5 < 0 \) implies \( x < -5 \)

Thus, for this case, \( x < -5 \).

Combining both cases, we have:
\[ x > 1 \quad \text{or} \quad x < -5 \]

### Step 2: Solve the inequality \( \log_{\frac{1}{2}}\left(\frac{x-1}{x+5}\right) \geq 0 \)

The logarithm \( \log_{\frac{1}{2}}(y) \) is non-negative when \( y \leq 1 \) (since the base \( \frac{1}{2} < 1 \)). Therefore, we need to solve:
\[ \frac{x-1}{x+5} \leq 1 \]

Rearranging gives:
\[ x - 1 \leq x + 5 \]
This simplifies to:
\[ -1 \leq 5 \quad \text{(which is always true)} \]

Thus, there are no additional restrictions from this inequality.

### Step 3: Combine the results

From Step 1, we have two intervals:
1. \( x > 1 \)
2. \( x < -5 \)

Since the second condition does not impose any additional restrictions, the final solution for the values of \( x \) for which \( f(x) \) is defined is:
\[ x > 1 \quad \text{or} \quad x < -5 \]

### Summary of the Solution
The function \( f(x) \) is defined for:
\[ x \in (-\infty, -5) \cup (1, \infty) \]