find domain of composite f(g(x)) = f(x)= $$ \sqrt{8-x} $$ $$ \text{ }g\text{ }=\text{ }\frac{1}{x-3} $$

To find the domain of the composite function $ f(g(x)) $, where:

- $ f(x) = \sqrt{8 - x} $
- $ g(x) = \frac{1}{x - 3} $

### Step 1: Domain of $ f(x) = \sqrt{8 - x} $
For the square root function to be defined, the expression inside the square root must be non-negative. Thus, we require:

\[
8 - x \geq 0
\]

\[
x \leq 8
\]

So, the domain of $ f(x) $ is $ (-\infty, 8] $.

### Step 2: Domain of $ g(x) = \frac{1}{x - 3} $
The function $ g(x) $ is undefined when the denominator is zero, so we exclude $ x = 3 $. Therefore, the domain of $ g(x) $ is:

\[
x \in (-\infty, 3) \cup (3, \infty)
\]

### Step 3: Domain of the Composite Function $ f(g(x)) $
Now, we evaluate the domain of $ f(g(x)) $. First, for $ f(g(x)) $ to be defined, $ g(x) $ must lie in the domain of $ f(x) $, which is $ (-\infty, 8] $.

Since $ g(x) = \frac{1}{x - 3} $, we require:

\[
g(x) = \frac{1}{x - 3} \leq 8
\]

Solving this inequality:

\[
\frac{1}{x - 3} \leq 8
\]

Multiply both sides by $ (x - 3)^2 $ (which is positive for $ x \neq 3 $):

\[
1 \leq 8(x - 3)
\]

\[
1 \leq 8x - 24
\]

\[
25 \leq 8x
\]

\[
x \geq \frac{25}{8}
\]

Thus, we require $ x \geq \frac{25}{8} $ and $ x \neq 3 $ (since $ g(x) $ is undefined at $ x = 3 $).

### Final Answer: Domain of $ f(g(x)) $
The domain of the composite function $ f(g(x)) $ is:

\[
x \in \left[ \frac{25}{8}, 3 \right) \cup (3, \infty)
\]

Would you like more details on any part of the solution? Here are some related questions:

1. How do we determine the range of a composite function?
2. What happens when $ f $ or $ g $ are rational functions?
3. Can composite functions have a restricted range as well?
4. How do you simplify the inequality in a composite function's domain?
5. Why does the function $ g(x) = \frac{1}{x - 3} $ create a discontinuity at $ x = 3 $?

**Tip:** When solving inequalities involving fractions, remember to consider both positive and negative cases separately if the expression contains a variable in the denominator.

find the domain of composite f(g(x)) = f(x)= $ \sqrt{8-x} $ $ g(x) = \frac{1}{x-3} $

This detailed solution shows how to find the domain of the composite function f(g(x)), where f(x) = √(8 - x) and g(x) = 1/(x - 3). Learn how to evaluate the domain of both the square root and rational functions and apply inequalities to determine the valid input values. Suitable for Grades 10-12.

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