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We are asked to find the domain of the composite function \( f \circ g \), where:
- \( f(x) = \frac{5}{x - 7} \)
- \( g(x) = \sqrt{x - 5} \)

The domain of a composite function \( f(g(x)) \) depends on two factors:
1. The domain of \( g(x) \), which must be valid for the input to \( g \).
2. The domain of \( f(g(x)) \), which must also be valid for the result from \( g(x) \).

### Step 1: Domain of \( g(x) = \sqrt{x - 5} \)
The square root function is defined only for non-negative numbers, so:
\[
x - 5 \geq 0 \quad \Rightarrow \quad x \geq 5
\]
Thus, the domain of \( g(x) \) is \( x \geq 5 \).

### Step 2: Domain of \( f(x) = \frac{5}{x - 7} \)
The function \( f(x) \) is undefined when the denominator is zero, so:
\[
x - 7 \neq 0 \quad \Rightarrow \quad x \neq 7
\]
Therefore, \( f(x) \) is defined for all \( x \) except \( x = 7 \).

### Step 3: Domain of the Composite Function \( f \circ g \)
Now, we substitute \( g(x) \) into \( f \). This gives:
\[
f(g(x)) = \frac{5}{g(x) - 7} = \frac{5}{\sqrt{x - 5} - 7}
\]
We already know that \( x \geq 5 \), and now we must also exclude any \( x \) such that \( \sqrt{x - 5} - 7 = 0 \):
\[
\sqrt{x - 5} = 7 \quad \Rightarrow \quad x - 5 = 49 \quad \Rightarrow \quad x = 54
\]
Thus, we exclude \( x = 54 \) from the domain.

### Final Domain:
The domain of \( f \circ g \) is \( x \geq 5 \) but excluding \( x = 7 \) and \( x = 54 \).

So the correct answer is:
\[
\boxed{\text{d. } \{x \, | \, x \geq 5, \, x \neq 7, \, x \neq 54\}}
\]

Would you like further details on any step?

Here are five related questions to practice:
1. What is the domain of \( f(x) = \frac{3}{x^2 - 9} \)?
2. How do you find the range of a composite function?
3. What is the domain of \( g(x) = \sqrt{2x + 4} \)?
4. How do you handle square root functions when finding the domain?
5. How does adding absolute value symbols to a function affect its domain?

**Tip:** Always check the domain restrictions for both the inner and outer functions when finding the domain of a composite function.

Find the domain of the composite function f∘g, where f(x) = 5 / (x - 7) and g(x) = √(x - 5).

This problem explores how to find the domain of the composite function f∘g, involving a square root function and a rational function. Detailed steps guide you through identifying the domain restrictions for both g(x) and f(g(x)). Suitable for high school students in Grades 10-12.

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