Question

question 3 of 10 if $f(x) = 3x^2$ and $g(x) = x + 2$, find $(f \bullet g)(x)$. \\(\bigcirc\\) a. $3x^2 + 6x$ \\(\bigcirc\\) b. $x^3 + 2$ \\(\bigcirc\\) c. $3x^3 + 6x$ \\(\bigcirc\\) d. $3x^3 + 6x^2$

Explanation:

Step1: Recall the definition of function multiplication

The product of two functions \( (f \cdot g)(x) \) is defined as \( f(x) \cdot g(x) \).

Step2: Substitute the given functions

Given \( f(x) = 3x^2 \) and \( g(x) = x + 2 \), we substitute these into the product formula:
\( (f \cdot g)(x) = f(x) \cdot g(x) = 3x^2 \cdot (x + 2) \)

Step3: Distribute the \( 3x^2 \)

Using the distributive property \( a(b + c) = ab + ac \), where \( a = 3x^2 \), \( b = x \), and \( c = 2 \):
\( 3x^2 \cdot x + 3x^2 \cdot 2 \)

Step4: Simplify the terms

For the first term, \( 3x^2 \cdot x = 3x^{2 + 1} = 3x^3 \) (using the rule \( x^m \cdot x^n = x^{m + n} \)).
For the second term, \( 3x^2 \cdot 2 = 6x^2 \).
So, \( (f \cdot g)(x) = 3x^3 + 6x^2 \).

Answer:

D. \( 3x^3 + 6x^2 \)