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# Find $(f \cdot g)(-2)$ for the following functions.$\begin{array}{l}{f(x)=-x^{2}+2 x-2} \\ {g(x)=2 x+3}\end{array}$A. -20 B. 8 C. 120 D. 215

Solution

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By definition: $(f\cdot g)(x)=f(x)\cdot g(x)$.

First, we find $(f\cdot g)(x)$.

\begin{align*} (f\cdot g)(x)&=f(x)\cdot g(x)\\ &=(-x^2+2x-2)\cdot (2x+3)\\ &=-2x^3-3x^2+4x^2+6x-4x-6\\ &=-2x^3+x^2+2x-6 \end{align*}

Next, we replace the variable $x$ with the value $-2$.

\begin{align*} (f\cdot g)(-2)&=-2\cdot(-2)^3+(-2)^2+2\cdot(-2)-6\\ &=-16+4-4-6\\ &=\boxed{10} \end{align*}

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