Question
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
Real Tutor Solution
Tutor-Verified Answer
Answer
\((f \cdot g)(x) = -4x^2 - 5x + 6\)
Solution
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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Simplify this solution 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: \[ (f \cdot g)(x) = -4x^2 - 5x + 6 \]
