To determine [tex]\((f \cdot g)(x)\)[/tex], we need to find the product of the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. We start by expressing the given functions:
[tex]\[ f(x) = x - 3 \][/tex]
[tex]\[ g(x) = -2x + 4 \][/tex]
To find [tex]\((f \cdot g)(x)\)[/tex], we multiply [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x) = (x - 3) \cdot (-2x + 4)
\][/tex]
Next, we apply the distributive property to expand the expression:
[tex]\[
(x - 3)(-2x + 4) = x \cdot (-2x) + x \cdot 4 - 3 \cdot (-2x) - 3 \cdot 4
\][/tex]
Calculate each term:
[tex]\[
x \cdot (-2x) = -2x^2
\][/tex]
[tex]\[
x \cdot 4 = 4x
\][/tex]
[tex]\[
-3 \cdot (-2x) = 6x
\][/tex]
[tex]\[
-3 \cdot 4 = -12
\][/tex]
Now, add all the terms together:
[tex]\[
(f \cdot g)(x) = -2x^2 + 4x + 6x - 12
\][/tex]
Combine like terms:
[tex]\[
(f \cdot g)(x) = -2x^2 + (4x + 6x) - 12 = -2x^2 + 10x - 12
\][/tex]
Thus, the simplified expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
(f \cdot g)(x) = -2x^2 + 10x - 12
\][/tex]
Therefore, the correct answer is:
[tex]\((f \cdot g)(x) = -2x^2 + 10x - 12\)[/tex]
