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The image contains a mathematical problem related to function composition. Let's break down and solve each part:

Given:
- \( f(x) = x^2 + 4 \)
- \( g(x) = x^2 - 3 \)

You are asked to find:

### a. \( (f \circ g)(x) \) 
This means \( f(g(x)) \).

### b. \( (g \circ f)(x) \) 
This means \( g(f(x)) \).

### c. \( (f \circ g)(4) \) 
This means \( f(g(4)) \).

### d. \( (g \circ f)(4) \) 
This means \( g(f(4)) \).

Let's calculate each one step by step.

### a. \( (f \circ g)(x) \)
We substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(x^2 - 3) = (x^2 - 3)^2 + 4
\]
Now, simplify:
\[
(x^2 - 3)^2 = x^4 - 6x^2 + 9
\]
So:
\[
f(g(x)) = x^4 - 6x^2 + 9 + 4 = x^4 - 6x^2 + 13
\]

### b. \( (g \circ f)(x) \)
We substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(x^2 + 4) = (x^2 + 4)^2 - 3
\]
Now, simplify:
\[
(x^2 + 4)^2 = x^4 + 8x^2 + 16
\]
So:
\[
g(f(x)) = x^4 + 8x^2 + 16 - 3 = x^4 + 8x^2 + 13
\]

### c. \( (f \circ g)(4) \)
First, calculate \( g(4) \):
\[
g(4) = 4^2 - 3 = 16 - 3 = 13
\]
Now, substitute into \( f \):
\[
f(13) = 13^2 + 4 = 169 + 4 = 173
\]

### d. \( (g \circ f)(4) \)
First, calculate \( f(4) \):
\[
f(4) = 4^2 + 4 = 16 + 4 = 20
\]
Now, substitute into \( g \):
\[
g(20) = 20^2 - 3 = 400 - 3 = 397
\]

### Final Answers:
- \( (f \circ g)(x) = x^4 - 6x^2 + 13 \)
- \( (g \circ f)(x) = x^4 + 8x^2 + 13 \)
- \( (f \circ g)(4) = 173 \)
- \( (g \circ f)(4) = 397 \)

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions to extend this topic:
1. How do you prove the general composition of two functions \( f(g(x)) \)?
2. What are some practical applications of function composition?
3. Can you express \( f \circ g \) and \( g \circ f \) graphically?
4. How do inverses of functions relate to function composition?
5. How does the order of function composition affect the result?

**Tip:** Always remember that the order of composition matters—\( f(g(x)) \) is generally different from \( g(f(x)) \).

For f(x) = x^2 + 4 and g(x) = x^2 - 3, find the following functions:
a. (f ◦ g)(x);
b. (g ◦ f)(x);
c. (f ◦ g)(4);
d. (g ◦ f)(4).

Learn how to solve function compositions involving f(x) = x^2 + 4 and g(x) = x^2 - 3. This problem covers the key concepts of function composition, including calculating (f ◦ g)(x), (g ◦ f)(x), and evaluating at specific points like (f ◦ g)(4) and (g ◦ f)(4). Suitable for high school students.

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