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Answer :
To find [tex]\((f \cdot g)(x)\)[/tex], we'll start by considering what it means to have this product. Given two functions, [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], the product [tex]\((f \cdot g)(x)\)[/tex] is simply the function [tex]\(f(x)\)[/tex] multiplied by the function [tex]\(g(x)\)[/tex].
Let's break this down step-by-step:
1. Define the functions:
- [tex]\(f(x) = -5x\)[/tex]
- [tex]\(g(x) = 8x^2 - 5x - 9\)[/tex]
2. Calculate the product [tex]\((f \cdot g)(x)\)[/tex]:
- To find [tex]\((f \cdot g)(x)\)[/tex], you multiply [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x) = (-5x) \cdot (8x^2 - 5x - 9)
\][/tex]
3. Distribute to simplify:
- Distribute [tex]\( -5x \)[/tex] across each term in [tex]\( g(x) \)[/tex]:
[tex]\[
(f \cdot g)(x) = -5x \cdot 8x^2 + (-5x) \cdot (-5x) + (-5x) \cdot (-9)
\][/tex]
- Compute each multiplication:
- [tex]\( -5x \cdot 8x^2 = -40x^3 \)[/tex]
- [tex]\( -5x \cdot (-5x) = 25x^2 \)[/tex]
- [tex]\( -5x \cdot (-9) = 45x \)[/tex]
4. Combine the results to write the final product:
[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]
This expression, [tex]\(-40x^3 + 25x^2 + 45x\)[/tex], represents the product of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Let's break this down step-by-step:
1. Define the functions:
- [tex]\(f(x) = -5x\)[/tex]
- [tex]\(g(x) = 8x^2 - 5x - 9\)[/tex]
2. Calculate the product [tex]\((f \cdot g)(x)\)[/tex]:
- To find [tex]\((f \cdot g)(x)\)[/tex], you multiply [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x) = (-5x) \cdot (8x^2 - 5x - 9)
\][/tex]
3. Distribute to simplify:
- Distribute [tex]\( -5x \)[/tex] across each term in [tex]\( g(x) \)[/tex]:
[tex]\[
(f \cdot g)(x) = -5x \cdot 8x^2 + (-5x) \cdot (-5x) + (-5x) \cdot (-9)
\][/tex]
- Compute each multiplication:
- [tex]\( -5x \cdot 8x^2 = -40x^3 \)[/tex]
- [tex]\( -5x \cdot (-5x) = 25x^2 \)[/tex]
- [tex]\( -5x \cdot (-9) = 45x \)[/tex]
4. Combine the results to write the final product:
[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]
This expression, [tex]\(-40x^3 + 25x^2 + 45x\)[/tex], represents the product of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
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