The problem involves finding the composition of two functions, $f(x)$ and $g(x)$.

Given:

$f(x) = 2x^2 + 8, \quad g(x) = 6x + 6$

We need to find:

(a) $(f \circ g)(x)$
(b) $(g \circ f)(x)$

(a) Finding $(f \circ g)(x)$

The composition $(f \circ g)(x)$ means we substitute $g(x)$ into $f(x)$.

$(f \circ g)(x) = f(g(x)) = f(6x + 6)$

Now, substitute $6x + 6$ for $x$ in $f(x) = 2x^2 + 8$:

$f(6x + 6) = 2(6x + 6)^2 + 8$

Calculate $(6x + 6)^2$:

$(6x + 6)^2 = 36x^2 + 72x + 36$

Now substitute it back:

$f(6x + 6) = 2(36x^2 + 72x + 36) + 8 = 72x^2 + 144x + 72 + 8 = 72x^2 + 144x + 80$

Thus,

$(f \circ g)(x) = 72x^2 + 144x + 80$

(b) Finding $(g \circ f)(x)$

The composition $(g \circ f)(x)$ means we substitute $f(x)$ into $g(x)$.

$(g \circ f)(x) = g(f(x)) = g(2x^2 + 8)$

Now, substitute $2x^2 + 8$ for $x$ in $g(x) = 6x + 6$:

$g(2x^2 + 8) = 6(2x^2 + 8) + 6$

Calculate:

$6(2x^2 + 8) = 12x^2 + 48$

So,

$g(2x^2 + 8) = 12x^2 + 48 + 6 = 12x^2 + 54$

Thus,

$(g \circ f)(x) = 12x^2 + 54$

Final Answers:

(a) $(f \circ g)(x) = 72x^2 + 144x + 80$
(b) $(g \circ f)(x) = 12x^2 + 54$

Would you like a more detailed explanation of any part?

Here are some related questions you might find interesting:

1. How do we find the domain of a composition of functions?
2. What happens to the composition of functions when one function is non-linear?
3. How can we graph the compositions of these functions to understand their behavior?
4. What is the difference between $f(g(x))$ and $g(f(x))$?
5. Can compositions of functions be commutative?

Tip: When composing functions, always remember to substitute the inner function completely into the outer function!