Use the pair of functions to find *f*(*g*(*x*)) and *g*(*f*(*x*))

. Simplify your answers.
 
*f*(*x*)=1/(*x*−6)

, *g*(*x*)=(8/*x)*+6

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. \( g(x) = \frac{8}{x} + 6 \)

---

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

---

### 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, \( g(f(x)) = 8x - 42 \).

---

### 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.

Use the pair of functions to find f(g(x)) and g(f(x)). Simplify your answers. f(x) = 1/(x - 6), g(x) = (8/x) + 6

Learn how to compose two functions, f(x) = 1/(x - 6) and g(x) = (8/x) + 6, to find f(g(x)) and g(f(x)). This problem includes a step-by-step breakdown of the solution with simplified results. Ideal for students in Grades 10-12 studying function composition and rational functions.

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