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The given function is:

\[
F(x) = \frac{\sqrt{x+2}}{x-2}
\]

### Domain:
The domain of a function is the set of all values of \( x \) for which the function is defined.

1. **For the square root**: 
   The expression inside the square root, \( x + 2 \), must be non-negative (since the square root of a negative number is not a real number). So we need:
   \[
   x + 2 \geq 0 \implies x \geq -2
   \]

2. **For the denominator**: 
   The denominator \( x - 2 \) cannot be zero because division by zero is undefined. So we require:
   \[
   x - 2 \neq 0 \implies x \neq 2
   \]

Combining these two conditions, the domain of \( F(x) \) is:
\[
x \geq -2 \quad \text{and} \quad x \neq 2
\]
In interval notation, the domain is:
\[
[-2, 2) \cup (2, \infty)
\]

### Range:
The range is the set of all possible output values of \( F(x) \). To determine the range, consider the behavior of the function:

- The numerator \( \sqrt{x+2} \) gives non-negative values (starting from 0 when \( x = -2 \) and increasing as \( x \) increases).
- The denominator \( x - 2 \) can be either positive or negative, depending on the value of \( x \).

As \( x \to -2^+ \), \( F(x) \to 0^+ \) (since \( \sqrt{x+2} \to 0 \)).
As \( x \to 2^- \), \( F(x) \to -\infty \).
As \( x \to 2^+ \), \( F(x) \to +\infty \).
As \( x \to \infty \), \( F(x) \to 0^+ \).

Thus, the range of the function is all real numbers except 0, which the function approaches but never reaches:
\[
(-\infty, 0) \cup (0, \infty)
\]

---

Would you like any additional details or have further questions?

Here are 5 related questions to explore this topic further:
1. What happens to the function \( F(x) \) as \( x \to 2 \) from both sides?
2. How does the presence of the square root affect the graph of the function?
3. Can you sketch the graph of \( F(x) \) based on its domain and range?
4. How do the domain and range change if the function were \( F(x) = \frac{x+2}{x-2} \) without the square root?
5. How would the domain and range change if the denominator were \( x^2 - 4 \) instead?

**Tip**: Always check for both square roots and denominators when determining the domain, as they impose different restrictions on the function.

Find the domain and range of F(x) = (sqrt(x + 2)) / (x - 2)

Learn how to find the domain and range of the function F(x) = (sqrt(x + 2)) / (x - 2). This problem involves restrictions from square roots and rational functions. Suitable for students in Grades 9-12.

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