To find the product of the given functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], where [tex]\( f(x) = 3x - 9 \)[/tex] and [tex]\( g(x) = 4x + 2 \)[/tex], we proceed as follows:
1. Express the Product Function [tex]\((f \cdot g)(x)\)[/tex]:
[tex]\[(f \cdot g)(x) = f(x) \cdot g(x)\][/tex]
2. Substitute the Given Functions:
[tex]\[f(x) = 3x - 9\][/tex]
[tex]\[g(x) = 4x + 2\][/tex]
Therefore,
[tex]\[(f \cdot g)(x) = (3x - 9)(4x + 2)\][/tex]
3. Apply the Distributive Property to Expand the Product:
[tex]\[
(3x - 9)(4x + 2) = 3x \cdot 4x + 3x \cdot 2 - 9 \cdot 4x - 9 \cdot 2
\][/tex]
4. Carry Out the Multiplications:
[tex]\[
3x \cdot 4x = 12x^2
\][/tex]
[tex]\[
3x \cdot 2 = 6x
\][/tex]
[tex]\[
-9 \cdot 4x = -36x
\][/tex]
[tex]\[
-9 \cdot 2 = -18
\][/tex]
5. Combine the Like Terms:
[tex]\[
12x^2 + 6x - 36x - 18
\][/tex]
[tex]\[
12x^2 - 30x - 18
\][/tex]
Therefore, the expression for [tex]\( (f \cdot g)(x) \)[/tex] is:
[tex]\[
(f \cdot g)(x) = 12x^2 - 30x - 18
\][/tex]
6. Determine the Domain of [tex]\( (f \cdot g)(x) \)[/tex]:
Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are polynomials, and the product of polynomials is also a polynomial. The domain of any polynomial is all real numbers.
Therefore, the domain of [tex]\( (f \cdot g)(x) \)[/tex] is all real numbers.
In conclusion, the correct expression and domain for [tex]\( (f \cdot g)(x) \)[/tex] are:
[tex]\[
\boxed{(f \cdot g)(x) = 12x^2 - 30x - 18 \text{; all real numbers}}
\][/tex]
Hence, the correct answer from the given options is:
[tex]\[
\boxed{D. (f \cdot g)(x) = 12x^2 - 30x - 18 \text{; all real numbers}}
\][/tex]
