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To find the domain of the composite function \( f \circ g \), we need to understand the domain restrictions for both functions \( f(x) \) and \( g(x) \) individually and then for the composition.

Given:
- \( f(x) = \frac{2}{x - 7} \)
- \( g(x) = \sqrt{x - 2} \)

1. **Domain of \( g(x) \):**  
   The square root function \( g(x) = \sqrt{x - 2} \) requires that \( x - 2 \geq 0 \).  
   Solving this inequality:
   \[
   x \geq 2
   \]
   So, the domain of \( g(x) \) is \( x \geq 2 \).

2. **Domain of \( f(g(x)) \):**  
   For the composite function \( f(g(x)) \), we need \( g(x) \neq 7 \) because of the denominator in \( f(x) \).  
   Setting \( g(x) = 7 \):
   \[
   \sqrt{x - 2} = 7
   \]
   Squaring both sides:
   \[
   x - 2 = 49 \Rightarrow x = 51
   \]
   Therefore, \( x \neq 51 \) to avoid division by zero in \( f(g(x)) \).

3. **Combined Domain:**  
   Combining these restrictions, we have:
   - \( x \geq 2 \)
   - \( x \neq 51 \)
   
   The correct answer is:
   \[
   \{ x \mid x \geq 2, x \neq 51 \}
   \]

### Answer
The correct option is:
- \( \{ x \mid x \geq 2, x \neq 51 \} \)

Would you like further details on the solution process?

Here are some related questions you might explore:
1. How do we determine the domain of a function involving a square root?
2. Why does \( x = 51 \) need to be excluded in this composite function?
3. How do restrictions in the inner function \( g(x) \) affect the domain of \( f(g(x)) \)?
4. What other types of functions require special attention for domain restrictions?
5. How would the answer change if \( f(x) \) had a different form?

**Tip:** When determining the domain of composite functions, always start with the inner function and work outward, applying any additional restrictions introduced by the outer function.

Find the domain of the composite function f ∘ g, where f(x) = 2 / (x - 7) and g(x) = √(x - 2).

Learn how to determine the domain of a composite function, given f(x) = 2 / (x - 7) and g(x) = √(x - 2). This solution covers the steps for handling domain restrictions in square root and rational functions, suitable for Grades 10-12.

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