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When dealing with a function like \(F(x)=\sqrt{\frac{x-5}{x+6}}\), it is vital to understand where the function is defined. Two primary domain restrictions need to be considered: the square root and the fraction.
1. **Square root condition**: The expression inside the square root must be non-negative because the square root of a negative number is not defined in the set of real numbers. This gives us the inequality \( \frac{x-5}{x+6} \geq 0 \).
2. **Fraction condition**: The denominator of a fraction cannot be zero, as division by zero is undefined. Therefore, we must ensure that \( x+6 eq 0 \), or \( x eq -6 \).
By combining these two conditions, we can start to figure out the domain of function.

To find where the fraction \( \frac{x-5}{x+6} \geq 0 \), we need to determine where the numerator and the denominator have the same sign. This can be done through solving the inequality:
1. The fraction will be non-negative when both the numerator \( x-5 \) and the denominator \( x+6 \) are either both positive or both negative.
2. Solve for when these conditions hold true:
- \( x-5 \geq 0 \Rightarrow x \geq 5 \).
- \( x+6 \geq 0 \Rightarrow x \geq -6 \).
Next, consider the opposite:
- \( x-5 \leq 0 \Rightarrow x \leq 5 \).
- \( x+6 \leq 0 \Rightarrow x \leq -6 \).
By solving these scenarios, we identify where the expression inside the square root remains non-negative.

Critical points are specific values of x where the function might change its behavior, mainly where the fraction changes sign. To find the critical points for \( \frac{x-5}{x+6} \):
1. **Find where the numerator is zero**: Set \( x-5=0 \Rightarrow x=5 \).
2. **Find where the denominator is zero**: Set \( x+6=0 \Rightarrow x=-6 \). These points separate the number line into different intervals which will be analyzed independently.\( x=-6 \) is particularly important since it makes the denominator zero, making the function undefined.

Once the critical points are identified, examine the three intervals created: \([-\text{∞}, -6)\), \((-6, 5)\), and \((5, \text{∞})\). Determine if \( \frac{x-5}{x+6} \geq 0 \) in each interval:
1. For the interval \([-\text{∞}, -6)\): Both the numerator \( x-5 \) and the denominator \( x+6 \) are negative, resulting in a positive fraction.
2. For the interval \((-6, 5)\): Here, \( x-5 \) is negative and \( x+6 \) is positive, leading to a negative fraction.
3. For the interval \((5, \text{∞})\): Both \( x-5 \) and \( x+6 \) are positive, resulting in a positive fraction. This analysis will help in understanding where the function is continuous.

Based on the previous intervals, we determine where the function \( F(x)=\sqrt{\frac{x-5}{x+6}} \) is defined and continuous:
1. The function is defined and non-negative at intervals where the fraction is positive: \([-\text{∞}, -6)\) and \((5, \text{∞})\).
2. Exclude the point \( x=-6 \), as it makes the denominator zero and hence the fraction is undefined at this point.
By combining these observations, we conclude that the function is continuous and defined on the intervals \([-\text{∞}, -6)\) and \((5, \text{∞})\). These intervals are where the function behaves smoothly without interruptions.