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Question Find (f⋅g)(x)(f \cdot g)(x) and (f⋅g)(1)(f \cdot g)(1) for f(x)=−7x+3f(x)=-7x+3 and g(x)=3x2−4x−1g(x)=3x^2-4x-1.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x) is given by f(x)=−7x+3f(x)=-7x+3 . The function g(x)g(x) is given by g(x)=3x−4x−1g(x)=3x^{}-4x-1
3. We are asked to find the product of the two functions, denoted as (f⋅g)(x)(f \cdot g)(x)4. We are also asked to evaluate this product at x=1x=1, denoted as (f⋅g)(1)(f \cdot g)(1)

STEP 2

First, we need to find the product of the two functions f(x)f(x) and g(x)g(x). This is done by multiplying the expressions for f(x)f(x) and g(x)g(x) together.
(f⋅g)(x)=f(x)⋅g(x)(f \cdot g)(x) = f(x) \cdot g(x)

STEP 3

Now, plug in the given expressions for f(x)f(x) and g(x)g(x).
(f⋅g)(x)=(−7x+3)⋅(3x2−x−1)(f \cdot g)(x) = (-7x+3) \cdot (3x^{2}-x-1)

STEP 4

Next, we need to distribute each term in the first expression to each term in the second expression. This is done using the distributive property of multiplication over addition.
(f⋅g)(x)=−7x⋅3x2+−7x⋅−4x+−7x⋅−1+3⋅3x2+3⋅−4x+3⋅−1(f \cdot g)(x) = -7x \cdot3x^{2} + -7x \cdot -4x + -7x \cdot -1 +3 \cdot3x^{2} +3 \cdot -4x +3 \cdot -1

STEP 5

Now, simplify each term.
(f⋅g)(x)=−21x3+28x2+7x+9x2−12x−3(f \cdot g)(x) = -21x^{3} +28x^{2} +7x +9x^{2} -12x -3

STEP 6

Combine like terms.
(f⋅g)(x)=−21x3+37x2−5x−3(f \cdot g)(x) = -21x^{3} +37x^{2} -5x -3

STEP 7

Now that we have the expression for (f⋅g)(x)(f \cdot g)(x), we can evaluate it at x=1x=1.
(f⋅g)(1)=−21(1)3+37(1)2−5(1)−3(f \cdot g)(1) = -21(1)^{3} +37(1)^{2} -5(1) -3

STEP 8

implify the expression.
(f⋅g)(1)=−21+37−5−3(f \cdot g)(1) = -21 +37 -5 -3

STEP 9

Calculate the value.
(f⋅g)()=8(f \cdot g)() =8So, (f⋅g)(x)=−21x3+37x2−5x−3(f \cdot g)(x) = -21x^{3} +37x^{2} -5x -3 and (f⋅g)()=8(f \cdot g)() =8.

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