We are tasked with evaluating $g(f(x))$, where the two given functions are:

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

Step 1: Substitute $f(x)$ into $g(x)$

We substitute $f(x) = \frac{1}{x} + 2$ into the function $g(x)$. This gives:

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

Substitute $f(x) = \frac{1}{x} + 2$:

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

Step 2: Simplify the numerator and denominator

Now, simplify both the numerator and denominator.

• Numerator: $\frac{1}{x} + 2$ (This remains as it is).

• Denominator: $\left(\frac{1}{x} + 2\right) - 8$. Simplify the constant part:

  $2 - 8 = -6$

  So, the denominator becomes:

  $\frac{1}{x} - 6$

Step 3: Write the final expression

Now, substitute these simplified expressions into the equation:

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

This is the final expression for $g(f(x))$.

Would you like me to simplify this further or evaluate for specific values of $x$?

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Here are 5 related questions to expand your understanding:

1. What happens when you evaluate $g(f(x))$ for a specific $x$?
2. How do you find the domain of the composite function $g(f(x))$?
3. What would $f(g(x))$ look like in this case?
4. How do vertical asymptotes change when combining two functions like this?
5. What is the behavior of $g(f(x))$ as $x$ approaches infinity?

Tip: When composing functions, always check for potential restrictions on the domain based on both functions involved.