Question

question 8 of 10
for $f(x) = 2x + 1$ and $g(x) = x^2 - 7$, find $(fcdot g)(x)$.

a. $3x^2 + 7x - 6$
b. $2x^2 - 13$
c. $2x^3 - x^2 - 7x - 7$
d. $2x^3 + x^2 - 14x - 7$

Explanation:

Step1: Recall the definition of function multiplication

To find \((f\cdot g)(x)\), we need to multiply the two functions \(f(x)\) and \(g(x)\) together. So, \((f\cdot g)(x)=f(x)\times g(x)\).
Given \(f(x) = 2x + 1\) and \(g(x)=x^{2}-7\), we substitute these into the formula: \((2x + 1)(x^{2}-7)\).

Step2: Expand the product using the distributive property (FOIL method for binomial and trinomial)

We distribute each term in the first binomial to each term in the second trinomial:
\[

$$\begin{align*} (2x + 1)(x^{2}-7)&=2x\times x^{2}+2x\times(-7)+1\times x^{2}+1\times(-7)\\ &=2x^{3}-14x+x^{2}-7\\ &=2x^{3}+x^{2}-14x - 7 \end{align*}$$

\]

Answer:

D. \(2x^{3}+x^{2}-14x - 7\)