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.