let $f(x) = x^2 + 3x$ and $g(x) = 5x^2 - 1$. find the function.

$f \cdot g$

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

find the domain. (enter your answer using interval notation.)

Question

let $f(x) = x^2 + 3x$ and $g(x) = 5x^2 - 1$. find the function.

$f \cdot g$

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

find the domain. (enter your answer using interval notation.)

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Interval Notation" ], "new_concepts": [ "Function Multiplication", "Domain of Product" ], "current_concepts": [ "Function Multiplication", "Domain of Product", "Interval Notation" ] } ## Multiply the functions $f(x)$ and $g(x)$ $$ \begin{aligned} (f \cdot g)(x) &= f(x) \cdot g(x) \ &= (x^2 + 3x)(5x^2 - 1) \ &= x^2(5x^2 - 1) + 3x(5x^2 - 1) \ &= 5x^4 - x^2 + 15x^3 - 3x \ &= 5x^4 + 15x^3 - x^2 - 3x \end{aligned} $$ ## Determine the domain of the product function $$ \begin{aligned} &\text{Since } f(x) \text{ and } g(x) \text{ are polynomials, their domains are } (-\infty, \infty).\ &\text{The domain of } (f \cdot g)(x) \text{ is the intersection of their domains: } (-\infty, \infty). \end{aligned} $$ ## Question 1 Let $f(x) = x^2 + 3x$ and $g(x) = 5x^2 - 1$. Find the function $f \cdot g$. $(f \cdot g)(x) =$ $5x^4 + 15x^3 - x^2 - 3x$ ## Question 2 Find the domain. (Enter your answer using interval notation.) $(-\infty, \infty)$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Function Multiplication" ] }

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Question

Explanation:

Multiply the functions \(f(x)\) and \(g(x)\)

Determine the domain of the product function

Answer:

Question 1

Question 2