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To find the domain of \( f(g(x)) \), we need to consider two factors:

1. The domain of \( g(x) \) itself.
2. The values of \( g(x) \) that make \( f(x) \) undefined.

Given:

\[
f(x) = \frac{1}{x+4} \quad \text{and} \quad g(x) = \frac{8}{x+1}
\]

### Step 1: Find the domain of \( g(x) \)
The function \( g(x) = \frac{8}{x+1} \) is undefined when the denominator is zero. So, set the denominator equal to zero and solve for \( x \):

\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]

Thus, the domain of \( g(x) \) is all real numbers except \( x = -1 \), or:

\[
\text{Domain of } g(x) = (-\infty, -1) \cup (-1, \infty)
\]

### Step 2: Find when \( f(g(x)) \) is undefined
Next, consider the function \( f(x) = \frac{1}{x+4} \), which is undefined when its denominator is zero. This happens when:

\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

For \( f(g(x)) \), we need to solve when \( g(x) = -4 \):

\[
\frac{8}{x+1} = -4
\]

Multiply both sides by \( x + 1 \):

\[
8 = -4(x + 1)
\]

Expand and solve for \( x \):

\[
8 = -4x - 4 \quad \Rightarrow \quad 12 = -4x \quad \Rightarrow \quad x = -3
\]

Thus, \( f(g(x)) \) is undefined at \( x = -3 \).

### Final Domain
The domain of \( f(g(x)) \) is all real numbers except where \( g(x) \) is undefined (at \( x = -1 \)) and where \( f(g(x)) \) is undefined (at \( x = -3 \)). Therefore, the domain of \( f(g(x)) \) is:

\[
\boxed{(-\infty, -3) \cup (-3, -1) \cup (-1, \infty)}
\]

Would you like more details or have any questions?

### Related Questions:
1. What is the domain of the individual functions \( f(x) \) and \( g(x) \)?
2. How do you find the composition of two functions?
3. What happens if both functions have restricted domains?
4. Can a function's range affect the domain of a composition of functions?
5. How can you express the domain of a function using interval notation?

#### Tip:
Always check both functions' restrictions when finding the domain of a composition.

Given f(x) = 1/(x + 4) and g(x) = 8/(x + 1), find the domain of f(g(x)).

Learn how to find the domain of the composed function f(g(x)) where f(x) = 1/(x + 4) and g(x) = 8/(x + 1). This solution involves key algebraic concepts such as function domains, rational functions, and composition of functions. Suitable for students in Grades 10-12.

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