Let 𝑓(𝑥)=1𝑥+2
 and 𝑔(𝑥)=𝑥𝑥−8
 . Evaluate the following: 𝑔(𝑓(𝑥))

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 \( x \)?

---

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.

Let 𝑓(𝑥)=\frac{1}{𝑥}+2 and 𝑔(𝑥)=\frac{𝑥}{𝑥−8}. Evaluate the following: 𝑔(𝑓(𝑥))

Explore how to evaluate the composite function g(f(x)) where f(x) = 1/x + 2 and g(x) = x/(x - 8). This problem covers key algebraic concepts such as function composition and rational expressions. Suitable for high school students in Grades 10-12.

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