Let's find $f(g(x))$ and $g(f(x))$ step by step, using the given functions.

Given:

1. $f(x) = \frac{1}{x - 6}$
2. $g(x) = \frac{8}{x} + 6$

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Finding $f(g(x))$:

We want to substitute $g(x)$ into $f(x)$. That is, we replace $x$ in $f(x)$ with $g(x)$.

$f(g(x)) = f\left( \frac{8}{x} + 6 \right) = \frac{1}{\left( \frac{8}{x} + 6 \right) - 6}$

Simplify the expression inside the parentheses:

$\frac{8}{x} + 6 - 6 = \frac{8}{x}$

So, now we have:

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

Thus, $f(g(x)) = \frac{x}{8}$.

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Finding $g(f(x))$:

We now substitute $f(x)$ into $g(x)$. That is, we replace $x$ in $g(x)$ with $f(x)$.

$g(f(x)) = g\left( \frac{1}{x - 6} \right) = \frac{8}{\frac{1}{x - 6}} + 6$

Simplify the first term:

$\frac{8}{\frac{1}{x - 6}} = 8(x - 6)$

Now, add 6:

$g(f(x)) = 8(x - 6) + 6 = 8x - 48 + 6 = 8x - 42$

Thus, $g(f(x)) = 8x - 42$.

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Final Answers:

1. $f(g(x)) = \frac{x}{8}$
2. $g(f(x)) = 8x - 42$

Would you like any further details on these steps?

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Here are 5 related questions to expand on this concept:

1. What happens if we reverse the order of composition for different functions?
2. How would the composition change if $f(x)$ or $g(x)$ were different?
3. Can we find the inverse of either function?
4. What would $f(g(x))$ look like if both functions were polynomials?
5. How do you interpret the composition of two functions geometrically?

Tip: When composing functions, always substitute the entire output of the inner function into the outer function's variable.