Pregunta
What is \( (f \cdot g)(x) \) ? \[ \begin{array}{l}f(x)=x+2 \\ g(x)=-4 x+3 \\ \text { Write your answer as a polynomial or a rational function in simplest form. }\end{array} . \begin{array}{l}\text { ? }\end{array} \text { ? } \]
Ask by Goodwin Vega. in the United States
Feb 03,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
\((f \cdot g)(x) = -4x^2 - 5x + 6\)
Solución
To find \((f \cdot g)(x)\), we multiply the two functions \(f(x)\) and \(g(x)\):
\[
\begin{align*}
f(x) &= x + 2 \\
g(x) &= -4x + 3 \\
(f \cdot g)(x) &= f(x) \cdot g(x) \\
&= (x + 2)(-4x + 3)
\end{align*}
\]
Now, expand the product:
\[
\begin{align*}
(f \cdot g)(x) &= x(-4x) + x(3) + 2(-4x) + 2(3) \\
&= -4x^2 + 3x - 8x + 6 \\
&= -4x^2 - 5x + 6
\end{align*}
\]
So, the simplified form of \((f \cdot g)(x)\) is:
\[
(f \cdot g)(x) = -4x^2 - 5x + 6
\]
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Simplifique esta solución Extra Insights
To find \( (f \cdot g)(x) \), we need to multiply the two functions \( f(x) \) and \( g(x) \). First, substitute the given functions: \[ f(x) = x + 2 \] \[ g(x) = -4x + 3 \] Now, multiply them together: \[ (f \cdot g)(x) = f(x) \cdot g(x) = (x + 2)(-4x + 3) \] Expanding this product using the distributive property (also known as the FOIL method for binomials): 1. First: \( x \cdot (-4x) = -4x^2 \) 2. Outside: \( x \cdot 3 = 3x \) 3. Inside: \( 2 \cdot (-4x) = -8x \) 4. Last: \( 2 \cdot 3 = 6 \) Combining these results: \[ -4x^2 + 3x - 8x + 6 = -4x^2 - 5x + 6 \] Therefore, the answer in simplest form is: \[ (f \cdot g)(x) = -4x^2 - 5x + 6 \]
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