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The problem asks to find compositions of the functions \( f(x) = x^2 + 8 \) and \( g(x) = x^2 - 9 \).

Let's go through the steps:

### (a) \( (f \circ g)(x) \)
This means applying \( g(x) \) inside \( f(x) \), i.e., \( f(g(x)) \).
- \( g(x) = x^2 - 9 \).
- Substitute \( g(x) \) into \( f(x) \):
  \[
  f(g(x)) = f(x^2 - 9) = (x^2 - 9)^2 + 8.
  \]
- Now, expand \( (x^2 - 9)^2 \):
  \[
  (x^2 - 9)^2 = x^4 - 18x^2 + 81.
  \]
- Adding 8:
  \[
  f(g(x)) = x^4 - 18x^2 + 81 + 8 = x^4 - 18x^2 + 89.
  \]

Thus, \( (f \circ g)(x) = x^4 - 18x^2 + 89 \).

### (b) \( (g \circ f)(x) \)
This means applying \( f(x) \) inside \( g(x) \), i.e., \( g(f(x)) \).
- \( f(x) = x^2 + 8 \).
- Substitute \( f(x) \) into \( g(x) \):
  \[
  g(f(x)) = g(x^2 + 8) = (x^2 + 8)^2 - 9.
  \]
- Expand \( (x^2 + 8)^2 \):
  \[
  (x^2 + 8)^2 = x^4 + 16x^2 + 64.
  \]
- Subtract 9:
  \[
  g(f(x)) = x^4 + 16x^2 + 64 - 9 = x^4 + 16x^2 + 55.
  \]

Thus, \( (g \circ f)(x) = x^4 + 16x^2 + 55 \).

### (c) \( (f \circ g)(2) \)
We already know \( f(g(x)) = x^4 - 18x^2 + 89 \), so let's substitute \( x = 2 \):
  \[
  f(g(2)) = 2^4 - 18(2^2) + 89 = 16 - 18(4) + 89 = 16 - 72 + 89 = 33.
  \]

Thus, \( (f \circ g)(2) = 33 \).

### (d) \( (g \circ f)(2) \)
We know \( g(f(x)) = x^4 + 16x^2 + 55 \), so let's substitute \( x = 2 \):
  \[
  g(f(2)) = 2^4 + 16(2^2) + 55 = 16 + 16(4) + 55 = 16 + 64 + 55 = 135.
  \]

Thus, \( (g \circ f)(2) = 135 \).

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Would you like further details or have any questions?

Here are some related questions for you to explore:
1. What is the general form of a composition of two functions?
2. How does the order of composition affect the result?
3. How would you handle the composition of three functions?
4. Can you think of a scenario where compositions are used in real-world problems?
5. What changes in the result if one of the functions is linear instead of quadratic?

**Tip:** When composing functions, always follow the order carefully—it's crucial in both substitution and simplification.

For f(x) = x^2 + 8 and g(x) = x^2 - 9, find the following functions: (a) (f ∘ g)(x), (b) (g ∘ f)(x), (c) (f ∘ g)(2), and (d) (g ∘ f)(2).

Learn how to solve the composition of functions f(x) = x^2 + 8 and g(x) = x^2 - 9. This problem involves finding (f ∘ g)(x), (g ∘ f)(x), (f ∘ g)(2), and (g ∘ f)(2) with step-by-step solutions. Suitable for students in Grades 10-12.

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