Thank you for the clarification! Let's correct the problem using \( F(x) \), \( G(x) \), and \( H(x) \) where the functions are given as:

- \( F(x) = x^2 - 3 \)
- \( G(x) = x + 2 \)
- \( H(x) = \frac{1}{x - 5} \)

The problem asks to simplify the expression \( G(F(H(x))) \).

Let's proceed with the corrected steps:

1. **Find \( H(x) \):**
   \[
   H(x) = \frac{1}{x - 5}
   \]

2. **Substitute \( H(x) \) into \( F(x) \) to get \( F(H(x)) \):**
   \[
   F(H(x)) = F\left(\frac{1}{x - 5}\right) = \left(\frac{1}{x - 5}\right)^2 - 3
   \]
   Simplifying \( F(H(x)) \):
   \[
   F(H(x)) = \frac{1}{(x - 5)^2} - 3
   \]

3. **Substitute \( F(H(x)) \) into \( G(x) \) to get \( G(F(H(x))) \):**
   \[
   G(F(H(x))) = G\left(\frac{1}{(x - 5)^2} - 3\right)
   \]
   Since \( G(x) = x + 2 \), we substitute:
   \[
   G(F(H(x))) = \left(\frac{1}{(x - 5)^2} - 3\right) + 2
   \]
   Simplifying:
   \[
   G(F(H(x))) = \frac{1}{(x - 5)^2} - 1
   \]

So the correct simplified expression is:
\[
\boxed{\frac{1}{(x - 5)^2} - 1}
\]

Would you like more details or have any questions?

Here are some related questions to consider:
1. How does the domain of \( H(x) \) affect the domain of the final composition \( G(F(H(x))) \)?
2. How would the result change if \( G(x) \) were a different linear function?
3. Can you find the inverse of the function \( G(F(H(x))) \)?
4. What happens to the expression as \( x \) approaches 5?
5. How can you graphically represent the composition of these functions?

**Tip:** When composing functions, always pay close attention to each step of substitution and simplification to avoid errors.

Explore the composition of functions problem: G(F(H(x))) where F(x), G(x), and H(x) are defined functions. Step-by-step solution provided with detailed substitution and simplification. Suitable for Grades 11-12 students studying advanced function compositions in mathematics.

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