To find [tex]\((f \cdot g)(x)\)[/tex] given [tex]\(f(x) = 2x - 3\)[/tex] and [tex]\(g(x) = 3x - 1\)[/tex], we need to multiply these two functions together.
Let's go through the multiplication step-by-step:
1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 2x - 3
\][/tex]
[tex]\[
g(x) = 3x - 1
\][/tex]
2. To find the product [tex]\((f \cdot g)(x)\)[/tex], multiply the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = (2x - 3)(3x - 1)
\][/tex]
3. Use the distributive property (also known as FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[
(2x - 3)(3x - 1) = 2x \cdot 3x + 2x \cdot (-1) + (-3) \cdot 3x + (-3) \cdot (-1)
\][/tex]
4. Perform the multiplications:
[tex]\[
2x \cdot 3x = 6x^2
\][/tex]
[tex]\[
2x \cdot (-1) = -2x
\][/tex]
[tex]\[
(-3) \cdot 3x = -9x
\][/tex]
[tex]\[
(-3) \cdot (-1) = 3
\][/tex]
5. Combine all the terms:
[tex]\[
(f \cdot g)(x) = 6x^2 - 2x - 9x + 3
\][/tex]
6. Simplify by combining like terms ([tex]\(-2x\)[/tex] and [tex]\(-9x\)[/tex]):
[tex]\[
(f \cdot g)(x) = 6x^2 - 11x + 3
\][/tex]
Thus, the product of the functions [tex]\(f(x) = 2x - 3\)[/tex] and [tex]\(g(x) = 3x - 1\)[/tex] is:
[tex]\[
(f \cdot g)(x) = 6x^2 - 11x + 3
\][/tex]
The correct choice from the given options is:
[tex]\[
6x^2 - 11x + 3
\][/tex]
