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.