To find [tex]\((f \cdot g)(x)\)[/tex] where [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are given by:
[tex]\[ f(x) = -2x - 3 \][/tex]
[tex]\[ g(x) = 3x + 1 \][/tex]
We need to multiply these two functions together. This process involves using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
\][/tex]
Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = (-2x - 3) \cdot (3x + 1)
\][/tex]
Next, we distribute each term in the first binomial by each term in the second binomial:
[tex]\[
= (-2x \cdot 3x) + (-2x \cdot 1) + (-3 \cdot 3x) + (-3 \cdot 1)
\][/tex]
Now, calculate each term:
[tex]\[
= (-2x \cdot 3x) + (-2x \cdot 1) + (-3 \cdot 3x) + (-3 \cdot 1)
= -6x^2 - 2x - 9x - 3
\][/tex]
Combine the like terms:
[tex]\[
-6x^2 - 2x - 9x - 3 = -6x^2 - 11x - 3
\][/tex]
So the final result for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
(f \cdot g)(x) = -6x^2 - 11x - 3
\][/tex]
