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

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.