Express the function h(x)=(1)/(x-8) in the form f*g. If g(x)=(x-8), find the function f(x). Your answer is f(x)=

Given the function \( h(x) = \frac{1}{x-8} \) and \( g(x) = x-8 \), we need to express \( h(x) \) in the form \( (f \circ g)(x) = f(g(x)) \).

1. Identify \( h(x) \):
   \[
   h(x) = \frac{1}{x-8}
   \]

2. Given \( g(x) = x-8 \), we substitute \( g(x) \) into the function \( h(x) \) to find \( f(x) \):
   \[
   h(x) = f(g(x))
   \]

3. Since \( g(x) = x-8 \), substitute \( g(x) \) into \( h(x) \):
   \[
   h(x) = \frac{1}{g(x)}
   \]

4. Therefore, \( f(u) \) (where \( u = g(x) \)) is:
   \[
   f(u) = \frac{1}{u}
   \]

Thus, the function \( f(x) \) is:
\[
f(x) = \frac{1}{x}
\]

So the answer is:
\[
f(x) = \frac{1}{x}
\]

Would you like more details or have any questions about this solution?

Here are 8 related questions that expand on this information:
1. What is the domain of the function \( h(x) = \frac{1}{x-8} \)?
2. How do you find the inverse of \( h(x) = \frac{1}{x-8} \)?
3. If \( k(x) = 2(x-8) \), how would you express \( h(x) \) in the form \( f(k(x)) \)?
4. What is the range of the function \( f(x) = \frac{1}{x} \)?
5. How does the graph of \( h(x) = \frac{1}{x-8} \) compare to the graph of \( f(x) = \frac{1}{x} \)?
6. How do you find the asymptotes of the function \( h(x) = \frac{1}{x-8} \)?
7. If \( g(x) = (x-8)^2 \), how would you express \( h(x) \) in the form \( f(g(x)) \)?
8. What is the behavior of \( h(x) \) as \( x \) approaches 8?

**Tip:** When working with compositions of functions, always ensure to substitute carefully and check the domains of the functions involved.

Learn how to express the function h(x)=(1)/(x-8) in the form f*g by identifying the function f(x). Detailed step-by-step solution with substitution and domain considerations. Suitable for advanced high school students.

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