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We are tasked with finding \( f(g(x)) \) (composition of \( f \) and \( g \)) and \( g(f(x)) \) for the given functions in question 1:

### Given:
\( f(x) = x - 4 \)  
\( g(x) = 4 - x^2 \)

### 1. Finding \( f(g(x)) \):
To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \).

\[
f(g(x)) = f(4 - x^2) = (4 - x^2) - 4
\]
Simplifying:
\[
f(g(x)) = 4 - x^2 - 4 = -x^2
\]
So, \( f(g(x)) = -x^2 \).

### 2. Finding \( g(f(x)) \):
To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \).

\[
g(f(x)) = g(x - 4) = 4 - (x - 4)^2
\]
Now expand \( (x - 4)^2 \):
\[
(x - 4)^2 = x^2 - 8x + 16
\]
Thus:
\[
g(f(x)) = 4 - (x^2 - 8x + 16) = 4 - x^2 + 8x - 16
\]
Simplifying:
\[
g(f(x)) = -x^2 + 8x - 12
\]
So, \( g(f(x)) = -x^2 + 8x - 12 \).

### Summary:
- \( f(g(x)) = -x^2 \)
- \( g(f(x)) = -x^2 + 8x - 12 \)

Would you like further details or explanations?  
Here are 5 related questions for you to explore:

1. How do we interpret the composition of two functions geometrically?
2. What is the domain of \( f(g(x)) \) when the inner function has a restricted domain?
3. Can you derive the inverse of the function \( g(x) = 4 - x^2 \)?
4. How would you verify if two composed functions are inverses of each other?
5. What happens if the compositions \( f(g(x)) \) and \( g(f(x)) \) are the same function?

**Tip:** Always ensure that the domain of the composed function \( f(g(x)) \) is valid by considering the domain restrictions of both functions.

Solve question 1: Find f(g(x)) and g(f(x)) for f(x) = x - 4 and g(x) = 4 - x^2.

Learn how to find the composition of two functions, f(g(x)) and g(f(x)), for f(x) = x - 4 and g(x) = 4 - x^2. This solution covers function composition and quadratic function principles, suitable for high school students.

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