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Let's go through each problem one by one, solving the indicated operations.

### 1. \( g(x) = x^3 + 2x \), \( h(x) = x - 2 \)
Find \( (g \circ h)(5) \):
- First, find \( h(5) = 5 - 2 = 3 \).
- Now, find \( g(h(5)) = g(3) = 3^3 + 2(3) = 27 + 6 = 33 \).
- Answer: \( (g \circ h)(5) = 33 \).

### 2. \( h(x) = 2x + 4 \), \( g(x) = -x + 1 \)
Find \( (h \circ g)(-6) \):
- First, find \( g(-6) = -(-6) + 1 = 6 + 1 = 7 \).
- Now, find \( h(g(-6)) = h(7) = 2(7) + 4 = 14 + 4 = 18 \).
- Answer: \( (h \circ g)(-6) = 18 \).

### 3. \( g(x) = 3x + 3 \), \( f(x) = 3x - 4 \)
Find \( g(f(-7)) \):
- First, find \( f(-7) = 3(-7) - 4 = -21 - 4 = -25 \).
- Now, find \( g(f(-7)) = g(-25) = 3(-25) + 3 = -75 + 3 = -72 \).
- Answer: \( g(f(-7)) = -72 \).

### 4. \( g(x) = -x^3 + 5x^2 \), \( f(x) = 2x + 4 \)
Find \( g(f(-4)) \):
- First, find \( f(-4) = 2(-4) + 4 = -8 + 4 = -4 \).
- Now, find \( g(f(-4)) = g(-4) = -(-4)^3 + 5(-4)^2 = -(-64) + 5(16) = 64 + 80 = 144 \).
- Answer: \( g(f(-4)) = 144 \).

### 5. \( f(x) = x^3 - 4 \), \( g(x) = 4x + 4 \)
Find \( (f \circ g)(0) \):
- First, find \( g(0) = 4(0) + 4 = 4 \).
- Now, find \( f(g(0)) = f(4) = 4^3 - 4 = 64 - 4 = 60 \).
- Answer: \( (f \circ g)(0) = 60 \).

### 6. \( g(n) = 4n - 5 \), \( f(n) = n^3 - 5 \)
Find \( g(f(3)) \):
- First, find \( f(3) = 3^3 - 5 = 27 - 5 = 22 \).
- Now, find \( g(f(3)) = g(22) = 4(22) - 5 = 88 - 5 = 83 \).
- Answer: \( g(f(3)) = 83 \).

### 7. \( g(x) = 3x + 2 \), \( h(x) = x + 3 \)
Find \( g(h(-7)) \):
- First, find \( h(-7) = -7 + 3 = -4 \).
- Now, find \( g(h(-7)) = g(-4) = 3(-4) + 2 = -12 + 2 = -10 \).
- Answer: \( g(h(-7)) = -10 \).

### 8. \( h(t) = -t^3 - 5t^2 \), \( g(t) = -t - 5 \)
Find \( h(g(-5)) \):
- First, find \( g(-5) = -(-5) - 5 = 5 - 5 = 0 \).
- Now, find \( h(g(-5)) = h(0) = -(0)^3 - 5(0)^2 = 0 \).
- Answer: \( h(g(-5)) = 0 \).

### 9. \( f(t) = t^3 - 5t \), \( g(t) = 3t - 1 \)
Find \( f(g(-1)) \):
- First, find \( g(-1) = 3(-1) - 1 = -3 - 1 = -4 \).
- Now, find \( f(g(-1)) = f(-4) = (-4)^3 - 5(-4) = -64 + 20 = -44 \).
- Answer: \( f(g(-1)) = -44 \).

### 10. \( f(x) = 2x + 3 \), \( g(x) = x^2 + 5 \)
Find \( (f \circ g)(-4) \):
- First, find \( g(-4) = (-4)^2 + 5 = 16 + 5 = 21 \).
- Now, find \( f(g(-4)) = f(21) = 2(21) + 3 = 42 + 3 = 45 \).
- Answer: \( (f \circ g)(-4) = 45 \).

Let me know if you'd like more details on any of these problems or have any questions!

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**Related Questions**:
1. How do you find the domain of composite functions?
2. What is the significance of function composition in real-life applications?
3. Can every function be composed with another function?
4. How can we reverse a composition of functions?
5. What happens when you compose two linear functions?

**Tip**: Always evaluate the inner function first when performing function compositions.

Perform the indicated operation for the given functions (composite functions).

This page provides a detailed step-by-step solution to composite function problems such as (f ∘ g)(x), involving polynomial and linear functions. Suitable for high school students studying Algebra 2 concepts.

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