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.