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To solve for \( f(g(x)) \), let's first clarify the functions given:

1. \( f(x) = \frac{1}{x - 5} \)
2. \( g(x) = \frac{1}{x} - 5 \)

### Step 1: Substitute \( g(x) \) into \( f(x) \)

Since we want \( f(g(x)) \), we substitute \( g(x) \) into \( f \):
\[
f(g(x)) = f\left(\frac{1}{x} - 5\right)
\]

### Step 2: Substitute into \( f(x) = \frac{1}{x - 5} \)

Replace \( x \) in \( f(x) \) with \( g(x) = \frac{1}{x} - 5 \):
\[
f(g(x)) = \frac{1}{\left(\frac{1}{x} - 5\right) - 5}
\]

### Step 3: Simplify the Expression

Now simplify the expression inside the denominator:
\[
f(g(x)) = \frac{1}{\frac{1}{x} - 5 - 5}
\]
\[
= \frac{1}{\frac{1}{x} - 10}
\]

To further simplify:
\[
f(g(x)) = \frac{1}{\frac{1 - 10x}{x}} = \frac{x}{1 - 10x}
\]

So,
\[
f(g(x)) = \frac{x}{1 - 10x}
\]

### Answer:
\[
f(g(x)) = \frac{x}{1 - 10x}
\]

Would you like more details on the simplification steps? 

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Here are some related questions to explore further:
1. How would you calculate \( g(f(x)) \) instead of \( f(g(x)) \)?
2. What are the domains of \( f(x) \) and \( g(x) \), and how do they impact the domain of \( f(g(x)) \)?
3. How would you find \( f(f(x)) \) or \( g(g(x)) \)?
4. Can you determine the inverse functions for \( f(x) \) and \( g(x) \)?
5. What happens to \( f(g(x)) \) as \( x \) approaches certain values that make the function undefined?

**Tip:** When composing functions, always consider the domain restrictions that may affect the final result.

If f(x) = 1 / (x - 5) and g(x) = 1 / x - 5, find f(g(x)).

Explore how to find the composition f(g(x)) when f(x) = 1 / (x - 5) and g(x) = 1 / x - 5. This problem covers key concepts in function composition and rational expressions, suitable for high school students in Grades 10-12.

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