Question

1. the domain for f(x) and g(x) is the set of all real numbers. let f(x) = 3x + 5 and g(x) = x². find f(x) · g(x). options: x² - 3x - 5; 3x³ + 3x² + 5; 3x² + 5; 3x³ + 5x²

Explanation:

Step1: Substitute the functions

We know that \( f(x) = 3x + 5 \) and \( g(x)=x^2 \). To find \( f(x)\cdot g(x) \), we substitute the expressions of \( f(x) \) and \( g(x) \) into the product, so we get \( (3x + 5)\cdot x^2 \).

Step2: Distribute the multiplication

Using the distributive property of multiplication over addition (\( a(b + c)=ab+ac \)), we multiply \( x^2 \) with each term inside the parentheses:
\( (3x + 5)\cdot x^2=3x\cdot x^2+5\cdot x^2 \)

Step3: Simplify the exponents

Recall the exponent rule \( a^m\cdot a^n=a^{m + n} \). For \( 3x\cdot x^2 \), we have \( x\cdot x^2=x^{1 + 2}=x^3 \), so \( 3x\cdot x^2 = 3x^3 \), and \( 5\cdot x^2=5x^2 \). Combining these results, we get \( 3x^3+5x^2 \).

Answer:

\( 3x^3 + 5x^2 \) (corresponding to the option " \( 3x^3 + 5x^2 \) " among the given choices)