Answer :
To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together. Let's start by writing down these functions:
- [tex]\(f(x) = x + 4\)[/tex]
- [tex]\(g(x) = 3x^2 - 7\)[/tex]
The product [tex]\((f \cdot g)(x)\)[/tex] is given by:
[tex]\[
(f \cdot g)(x) = f(x) \times g(x)
\][/tex]
Substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we get:
[tex]\[
(f \cdot g)(x) = (x + 4)(3x^2 - 7)
\][/tex]
Now, let's expand this expression by distributing each term in [tex]\(x + 4\)[/tex] with each term in [tex]\(3x^2 - 7\)[/tex].
1. Multiply the [tex]\(x\)[/tex] in [tex]\(x + 4\)[/tex] with each term in [tex]\(3x^2 - 7\)[/tex]:
[tex]\[
x \times 3x^2 = 3x^3
\][/tex]
[tex]\[
x \times (-7) = -7x
\][/tex]
2. Multiply the [tex]\(4\)[/tex] in [tex]\(x + 4\)[/tex] with each term in [tex]\(3x^2 - 7\)[/tex]:
[tex]\[
4 \times 3x^2 = 12x^2
\][/tex]
[tex]\[
4 \times (-7) = -28
\][/tex]
Now, combine all these products:
[tex]\[
3x^3 + 12x^2 - 7x - 28
\][/tex]
Therefore, the expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
3x^3 + 12x^2 - 7x - 28
\][/tex]
So, the correct choice is C: [tex]\( \boxed{3x^3 + 12x^2 - 7x - 28} \)[/tex].
- [tex]\(f(x) = x + 4\)[/tex]
- [tex]\(g(x) = 3x^2 - 7\)[/tex]
The product [tex]\((f \cdot g)(x)\)[/tex] is given by:
[tex]\[
(f \cdot g)(x) = f(x) \times g(x)
\][/tex]
Substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we get:
[tex]\[
(f \cdot g)(x) = (x + 4)(3x^2 - 7)
\][/tex]
Now, let's expand this expression by distributing each term in [tex]\(x + 4\)[/tex] with each term in [tex]\(3x^2 - 7\)[/tex].
1. Multiply the [tex]\(x\)[/tex] in [tex]\(x + 4\)[/tex] with each term in [tex]\(3x^2 - 7\)[/tex]:
[tex]\[
x \times 3x^2 = 3x^3
\][/tex]
[tex]\[
x \times (-7) = -7x
\][/tex]
2. Multiply the [tex]\(4\)[/tex] in [tex]\(x + 4\)[/tex] with each term in [tex]\(3x^2 - 7\)[/tex]:
[tex]\[
4 \times 3x^2 = 12x^2
\][/tex]
[tex]\[
4 \times (-7) = -28
\][/tex]
Now, combine all these products:
[tex]\[
3x^3 + 12x^2 - 7x - 28
\][/tex]
Therefore, the expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
3x^3 + 12x^2 - 7x - 28
\][/tex]
So, the correct choice is C: [tex]\( \boxed{3x^3 + 12x^2 - 7x - 28} \)[/tex].
