Let's work step by step to find $f(g(x))$ and determine its domain.

Step 1: Define the functions

You are given:

• $f(x) = \frac{1}{x+3}$
• $g(x) = \frac{1}{x-9}$

Now, we need to find $f(g(x))$.

Step 2: Find $f(g(x))$

To find $f(g(x))$, substitute $g(x)$ into the function $f(x)$.

$f(g(x)) = f\left( \frac{1}{x-9} \right)$

Since $f(x) = \frac{1}{x+3}$, replace $x$ in $f(x)$ with $g(x) = \frac{1}{x-9}$:

$f(g(x)) = \frac{1}{\frac{1}{x-9} + 3}$

Step 3: Simplify the expression

To simplify $f(g(x))$, first combine the terms in the denominator:

$f(g(x)) = \frac{1}{\frac{1}{x-9} + 3} = \frac{1}{\frac{1 + 3(x-9)}{x-9}} = \frac{1}{\frac{1 + 3x - 27}{x-9}}$

This simplifies to:

$f(g(x)) = \frac{x-9}{1 + 3x - 27} = \frac{x-9}{3x - 26}$

Thus, the composite function is:

$f(g(x)) = \frac{x-9}{3x-26}$

Step 4: Find the domain of $f(g(x))$

The domain of a composite function $f(g(x))$ is determined by two things:

1. The domain of $g(x)$ itself.
2. The values of $x$ for which $f(g(x))$ is defined.

1. Domain of $g(x)$:

$g(x) = \frac{1}{x-9}$ is undefined when $x = 9$, because division by zero is undefined. So, $x \neq 9$.

2. Domain of $f(g(x))$:

The function $f(g(x)) = \frac{x-9}{3x-26}$ is undefined when the denominator is zero. So, solve:

$3x - 26 = 0$

$3x = 26 \quad \Rightarrow \quad x = \frac{26}{3}$

Therefore, $f(g(x))$ is undefined when $x = \frac{26}{3}$.

Step 5: Final domain

The domain of $f(g(x))$ is all real numbers except $x = 9$ and $x = \frac{26}{3}$. In interval notation, the domain is:

$\text{Domain of } f(g(x)) = (-\infty, 9) \cup (9, \frac{26}{3}) \cup (\frac{26}{3}, \infty)$

Final Answer:

$f(g(x)) = \frac{x-9}{3x-26}$ $\text{Domain: } (-\infty, 9) \cup (9, \frac{26}{3}) \cup (\frac{26}{3}, \infty)$

Would you like further details on any step?

Related Questions:

1. How do you find the inverse of a composite function?
2. What happens if $g(x)$ is a different rational function?
3. How can you graph the function $f(g(x))$ and its domain restrictions?
4. What is the difference between the domain of $f(x)$, $g(x)$, and $f(g(x))$?
5. How do you find the range of $f(g(x))$?

Tip: When working with rational functions, always check for values that make the denominator zero to determine the domain.