given that f(X)=1/x+3 and G(x) is 1/x-9, find f(g(x)) and it's domain

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.

Given that f(X)=1/x+3 and G(x) is 1/x-9, find f(g(x)) and its domain.

Learn how to find the composite function f(g(x)) where f(x) = 1/(x+3) and g(x) = 1/(x-9), and how to determine its domain. This problem involves rational functions, composite functions, and domain restrictions. Suitable for high school students in Grades 10-12.

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