College

Given:

[tex]
\[
\begin{array}{l}
f(x) = x + 4 \\
g(x) = 3x^2 - 7
\end{array}
\]
[/tex]

Find [tex](f \cdot g)(x)[/tex].

A. [tex](f \cdot g)(x) = 3x^3 + 28[/tex]

B. [tex](f \cdot g)(x) = 3x^3 + 12x^2 + 7x + 28[/tex]

C. [tex](f \cdot g)(x) = 3x^3 - 28[/tex]

D. [tex](f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28[/tex]

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], you need to multiply the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].

Given:
- [tex]\(f(x) = x + 4\)[/tex]
- [tex]\(g(x) = 3x^2 - 7\)[/tex]

Let's calculate [tex]\((f \cdot g)(x) = f(x) \times g(x)\)[/tex]:

1. Write out the multiplication:
[tex]\[
(f \cdot g)(x) = (x + 4)(3x^2 - 7)
\][/tex]

2. Distribute each term inside the parentheses:
- Multiply [tex]\(x\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
x \times 3x^2 = 3x^3
\][/tex]
[tex]\[
x \times (-7) = -7x
\][/tex]

- Multiply [tex]\(4\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
4 \times 3x^2 = 12x^2
\][/tex]
[tex]\[
4 \times (-7) = -28
\][/tex]

3. Combine all the terms:
[tex]\[
3x^3 + 12x^2 - 7x - 28
\][/tex]

The correct answer is: D. [tex]\((f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28\)[/tex].