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Answer :
To find [tex]\((f \cdot g)(x)\)[/tex] when given the functions [tex]\(f(x) = -5x\)[/tex] and [tex]\(g(x) = 8x^2 - 5x - 9\)[/tex], follow these steps:
1. Understand the operation: The notation [tex]\((f \cdot g)(x)\)[/tex] represents the multiplication of the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. This means we need to multiply the outputs of these functions.
2. Write the functions:
[tex]\[
f(x) = -5x
\][/tex]
[tex]\[
g(x) = 8x^2 - 5x - 9
\][/tex]
3. Multiply the functions:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
\][/tex]
Substitute [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into the equation:
[tex]\[
(f \cdot g)(x) = (-5x) \cdot (8x^2 - 5x - 9)
\][/tex]
4. Distribute [tex]\(-5x\)[/tex] to each term inside the parenthesis:
[tex]\[
(f \cdot g)(x) = -5x \cdot 8x^2 + -5x \cdot -5x + -5x \cdot -9
\][/tex]
5. Perform the multiplications:
[tex]\[
-5x \cdot 8x^2 = -40x^3
\][/tex]
[tex]\[
-5x \cdot -5x = 25x^2
\][/tex]
[tex]\[
-5x \cdot -9 = 45x
\][/tex]
6. Combine the results:
[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]
Thus, the correct expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]
Therefore, the correct answer is:
[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]
1. Understand the operation: The notation [tex]\((f \cdot g)(x)\)[/tex] represents the multiplication of the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. This means we need to multiply the outputs of these functions.
2. Write the functions:
[tex]\[
f(x) = -5x
\][/tex]
[tex]\[
g(x) = 8x^2 - 5x - 9
\][/tex]
3. Multiply the functions:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
\][/tex]
Substitute [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into the equation:
[tex]\[
(f \cdot g)(x) = (-5x) \cdot (8x^2 - 5x - 9)
\][/tex]
4. Distribute [tex]\(-5x\)[/tex] to each term inside the parenthesis:
[tex]\[
(f \cdot g)(x) = -5x \cdot 8x^2 + -5x \cdot -5x + -5x \cdot -9
\][/tex]
5. Perform the multiplications:
[tex]\[
-5x \cdot 8x^2 = -40x^3
\][/tex]
[tex]\[
-5x \cdot -5x = 25x^2
\][/tex]
[tex]\[
-5x \cdot -9 = 45x
\][/tex]
6. Combine the results:
[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]
Thus, the correct expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]
Therefore, the correct answer is:
[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]
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