College

Given the functions:

[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 + 12x^2 - 7x - 28[/tex]

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

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

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

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. The functions given are:

[tex]\[ f(x) = x + 4 \][/tex]

[tex]\[ g(x) = 3x^2 - 7 \][/tex]

The product [tex]\((f \cdot g)(x)\)[/tex] is calculated by multiplying [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:

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

To do this, we'll use the distributive property and distribute each term in [tex]\(f(x)\)[/tex] across each term in [tex]\(g(x)\)[/tex]:

1. 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]

2. 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]

Now, combine all the terms:

[tex]\[
3x^3 + 12x^2 - 7x - 28
\][/tex]

So, the expression for [tex]\((f \cdot g)(x)\)[/tex] is:

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

The correct answer is:

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