Question

question 6
1 part | -- of 1 point

for the given functions f and g, find \\(f \cdot g\\) and state its domain.

\\(f(x) = 3x^3 + 2\\); \\(g(x) = 5x^2 + 3\\)

a. \\((f \cdot g)(x) = 15x^5 + 9x^3 + 10x^2 + 6\\); \\(\\{x \mid x \
eq 0\\}\\)
b. \\((f \cdot g)(x) = 15x^5 + 9x^3 + 10x^2 + 6\\); all real numbers
c. \\((f \cdot g)(x) = 15x^6 + 9x^3 + 10x^2 + 6\\); all real numbers
d. \\((f \cdot g)(x) = 3x^3 + 5x^2 + 6\\); all real numbers

Explanation:

Define the function multiplication

Using the Algebra of Functions and Function Multiplication knowledge points

$$ (f \cdot g)(x) = f(x) \cdot g(x) $$

Substitute the given functions

Using the Function Multiplication knowledge point

$$ (f \cdot g)(x) = (3x^3 + 2)(5x^2 + 3) $$

Expand the polynomial product

Using the Function Multiplication knowledge point

$$
LATEXBLOCK0
$$

Determine the domain of the product function

Using the Algebra of Functions knowledge point
Since \(f(x)\) and \(g(x)\) are polynomials, their domains are all real numbers. The domain of their product is the intersection of their individual domains, which is also all real numbers.

Answer:

• A. \((f \cdot g)(x) = 15x^5 + 9x^3 + 10x^2 + 6\); \(\{x | x

eq 0\}\)

• B. \((f \cdot g)(x) = 15x^5 + 9x^3 + 10x^2 + 6\); all real numbers (Correct answer)
• C. \((f \cdot g)(x) = 15x^6 + 9x^3 + 10x^2 + 6\); all real numbers
• D. \((f \cdot g)(x) = 3x^3 + 5x^2 + 6\); all real numbers