We need to solve the given problem step-by-step.

The problem states:
Given $f(x) = x^2 - 3$ and $g(x) = \frac{x + 2}{x}$, find $g(f(4))$.

Solution:

1. Evaluate $f(4)$: $f(x) = x^2 - 3$ Substitute $x = 4$: $f(4) = 4^2 - 3 = 16 - 3 = 13$

   So, $f(4) = 13$.

2. Evaluate $g(f(4)) = g(13)$: The function $g(x) = \frac{x + 2}{x}$. Substitute $x = 13$: $g(13) = \frac{13 + 2}{13} = \frac{15}{13}$

Final Answer:

$\boxed{\frac{15}{13}}$

Let me know if you'd like further clarification or additional steps!

Questions to Expand Understanding:

1. What happens if $f(x)$ is modified to $x^2 + 3$ instead?
2. Can you describe how $g(x) = \frac{x+2}{x}$ behaves for very large or very small $x$?
3. How would you find $g(f(-4))$ using the same functions?
4. What is the domain of $g(x)$? Why is $x = 0$ excluded?
5. Could the composition $g(f(x))$ have any undefined values?

Tip:

Always simplify inner functions first when solving composite function problems. This ensures accuracy and reduces complexity.