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Let's break down the problem based on the image you uploaded.

We are given the following functions:

- \( f(x) = \frac{x}{x+1} \)
- \( g(x) = \frac{9}{x} \)

We are asked to find:
1. \( (f \circ g)(x) \), which is \( f(g(x)) \)
2. The domain of \( f \circ g \)

### Part (a): Finding \( (f \circ g)(x) \)

To compute \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f\left(\frac{9}{x}\right) = \frac{\frac{9}{x}}{\frac{9}{x} + 1}
\]

Now, simplify the expression:

\[
f(g(x)) = \frac{\frac{9}{x}}{\frac{9}{x} + 1} = \frac{\frac{9}{x}}{\frac{9 + x}{x}} = \frac{9}{9 + x}
\]

Thus, \( (f \circ g)(x) = \frac{9}{9 + x} \).

### Part (b): Finding the Domain of \( f \circ g \)

To find the domain of \( f \circ g \), we need to consider the domains of both functions:

1. **Domain of \( g(x) \)**: Since \( g(x) = \frac{9}{x} \), \( x \neq 0 \), so the domain of \( g(x) \) is \( (-\infty, 0) \cup (0, \infty) \).
   
2. **Domain of \( f(g(x)) \)**: For \( f(g(x)) = \frac{9}{9 + x} \), the expression \( 9 + x \neq 0 \), which gives \( x \neq -9 \).

So, the domain of \( f \circ g \) is all values of \( x \) where \( x \neq 0 \) and \( x \neq -9 \). Therefore, the domain is:

\[
(-\infty, -9) \cup (-9, 0) \cup (0, \infty)
\]

### Final Answers:
- \( (f \circ g)(x) = \frac{9}{9 + x} \)
- Domain: \( (-\infty, -9) \cup (-9, 0) \cup (0, \infty) \)

Would you like further details or have any questions? 

Here are 5 follow-up questions to expand on this:
1. What is the general process of finding the domain of a composition of functions?
2. How do you approach simplifying complex rational functions like \( f(g(x)) \)?
3. What would happen to the domain if the form of \( g(x) \) changed?
4. Can you apply this same method to other types of function compositions (e.g., logarithmic or trigonometric)?
5. How does the presence of an asymptote at \( x = 0 \) affect the domain?

**Tip**: Always check the domains of both functions when dealing with compositions to avoid missing important restrictions!

For f(x) = x / (x + 1) and g(x) = 9 / x, find (f o g)(x) and the domain of f o g.

Learn how to solve function composition problems with rational functions. This example uses f(x) = x / (x + 1) and g(x) = 9 / x to find (f o g)(x) and determine its domain. Explore key algebraic concepts like function composition and domain restriction.

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