To find the domain of the composition $f(g(x))$, let's go through each function and determine any restrictions.

Step 1: Identify the domains of $f(x)$ and $g(x)$.

1. Domain of $f(x) = \frac{1}{x - 9}$:

   • The function $f(x)$ is undefined when the denominator is zero.
   • So, $x - 9 = 0 \Rightarrow x = 9$.
   • Thus, the domain of $f(x)$ is all real numbers except $x = 9$.

2. Domain of $g(x) = \frac{7}{x} + 3$:

   • The function $g(x)$ is undefined when the denominator is zero.
   • So, $x = 0$.
   • Thus, the domain of $g(x)$ is all real numbers except $x = 0$.

Step 2: Find $f(g(x))$ and its domain.

The composition $f(g(x))$ involves substituting $g(x)$ into $f(x)$: $f(g(x)) = f\left(\frac{7}{x} + 3\right) = \frac{1}{\frac{7}{x} + 3 - 9}.$ Simplify the expression inside $f$: $f(g(x)) = \frac{1}{\frac{7}{x} - 6}.$

Now, for $f(g(x))$ to be defined, two conditions must be met:

1. $g(x)$ must be within the domain of $f(x)$, which excludes $g(x) = 9$.
2. $g(x)$ itself must be defined, which excludes $x = 0$.

Let's solve for when $g(x) = 9$: $\frac{7}{x} + 3 = 9.$ Subtract 3 from both sides: $\frac{7}{x} = 6.$ Multiply both sides by $x$: $7 = 6x.$ Solve for $x$: $x = \frac{7}{6}.$

Final Domain of $f(g(x))$:

The domain of $f(g(x))$ is all real numbers except $x = 0$ and $x = \frac{7}{6}$.

So, the answer is: $x = 0, \frac{7}{6}.$

Would you like further explanation or have any questions?

Here are some related questions you might find helpful:

1. What is the general approach to finding the domain of a composite function?
2. How do you handle restrictions in the inner function when finding a composite function's domain?
3. Why is it important to consider each function separately before finding the composite function?
4. How would the answer change if the functions had different forms or restrictions?
5. Can you explain why certain values make functions undefined in terms of real-world applications?

Tip: Always check for values that make any denominator zero or lead to undefined operations like square roots of negative numbers when determining a function's domain.