To find and simplify the product [tex]\((f \cdot g)(x)\)[/tex] of the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we follow these steps:
1. Define the functions:
[tex]\[
f(x) = -3x + 2
\][/tex]
[tex]\[
g(x) = -x + 1
\][/tex]
2. Form the product of the functions:
The product [tex]\((f \cdot g)(x)\)[/tex] is given by multiplying [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x) = (-3x + 2) \cdot (-x + 1)
\][/tex]
3. Expand the product:
Use the distributive property to expand the product:
[tex]\[
(-3x + 2) \cdot (-x + 1)
\][/tex]
Distribute each term in the first binomial by each term in the second binomial:
[tex]\[
= (-3x) \cdot (-x) + (-3x) \cdot 1 + 2 \cdot (-x) + 2 \cdot 1
\][/tex]
4. Calculate each term:
[tex]\[
= 3x^2 + (-3x) + (-2x) + 2
\][/tex]
5. Combine like terms:
Combine the [tex]\( -3x \)[/tex] and [tex]\( -2x \)[/tex] terms:
[tex]\[
= 3x^2 - 5x + 2
\][/tex]
Thus, the product [tex]\((f \cdot g)(x)\)[/tex] before and after simplification is:
[tex]\[
(f \cdot g)(x) = (1 - x)(2 - 3x)
\][/tex]
And when simplified:
[tex]\[
(f \cdot g)(x) = (x - 1)(3x - 2)
\][/tex]
Hence, the fully simplified expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
(f \cdot g)(x) = 3x^2 - 5x + 2
\][/tex]
