Solved on Oct 29, 2023

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

STEP 1

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

STEP 2

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

STEP 3

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

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.
$(f \cdot g)(x) = -7x \cdot3x^{2} + -7x \cdot -4x + -7x \cdot -1 +3 \cdot3x^{2} +3 \cdot -4x +3 \cdot -1$

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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

STEP 1

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

STEP 2

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

STEP 3

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

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.
$(f \cdot g)(x) = -7x \cdot3x^{2} + -7x \cdot -4x + -7x \cdot -1 +3 \cdot3x^{2} +3 \cdot -4x +3 \cdot -1$

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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

STEP 1

STEP 2

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

STEP 3

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

STEP 4

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(f⋅g)(x)=−21x3+28x2+7x+9x2−12x−3

STEP 6

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

STEP 7

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

STEP 9

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

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

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

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

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

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

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

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

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