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Let's start by addressing part **a** of the problem, which asks to find \( (f \circ g)(x) \), the composition of the functions \( f(x) \) and \( g(x) \).

Given:
- \( f(x) = x^2 + 8 \)
- \( g(x) = x^2 - 4 \)

The composition \( (f \circ g)(x) \) means we will substitute \( g(x) \) into \( f(x) \). In other words:

\[
(f \circ g)(x) = f(g(x))
\]

Now, substitute \( g(x) = x^2 - 4 \) into \( f(x) \):

\[
f(g(x)) = f(x^2 - 4)
\]

The function \( f(x) = x^2 + 8 \), so:

\[
f(x^2 - 4) = (x^2 - 4)^2 + 8
\]

Now, expand \( (x^2 - 4)^2 \):

\[
(x^2 - 4)^2 = x^4 - 8x^2 + 16
\]

Thus:

\[
f(g(x)) = x^4 - 8x^2 + 16 + 8 = x^4 - 8x^2 + 24
\]

So:

\[
(f \circ g)(x) = x^4 - 8x^2 + 24
\]

Would you like to proceed with parts **b**, **c**, and **d** of the problem, or do you have any questions about this solution?

Here are 5 questions related to this topic:

1. How do you compute the composition of two functions?
2. What is the general formula for expanding a binomial like \( (x^2 - 4)^2 \)?
3. How does function composition differ from regular multiplication of functions?
4. Can a function composition be reversible, i.e., is \( f(g(x)) = g(f(x)) \) always true?
5. How do you determine the domain of a composed function like \( (f \circ g)(x) \)?

**Tip:** When composing functions, always start by substituting the inner function into the outer function carefully, and simplify step by step.

For f(x) = x^2 + 8 and g(x) = x^2 - 4, find the following functions: a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(2); d. (g ∘ f)(2).

Learn how to solve the function composition problem with f(x) = x^2 + 8 and g(x) = x^2 - 4. Detailed step-by-step solutions provided for (f ∘ g)(x), (g ∘ f)(x), (f ∘ g)(2), and (g ∘ f)(2). Suitable for high school students in Grades 9-12.

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