Question

the domain for f(x) and g(x) is the set of all real numbers. let f(x) = x² + 1 and g(x) = 3x. find f·g. a. 3x³ + 3x b. x² + 3x + 1 c. 3x³ + 3x + 1 d. x² + 3x

Explanation:

Step1: Recall the product of functions

To find \( f \cdot g \), we need to multiply the functions \( f(x) \) and \( g(x) \). So, \( (f \cdot g)(x)=f(x)\times g(x) \).

Step2: Substitute the given functions

We know that \( f(x)=x^{2}+1 \) and \( g(x)=3x \). Substituting these into the product formula, we get:
\( (f \cdot g)(x)=(x^{2}+1)\times(3x) \)

Step3: Distribute the multiplication

Using the distributive property (also known as the distributive law of multiplication over addition), we multiply \( 3x \) with each term inside the parentheses:
\( (x^{2}+1)\times(3x)=x^{2}\times(3x)+1\times(3x) \)

Step4: Simplify each term

For the first term, \( x^{2}\times(3x) = 3x^{3} \) (using the rule of exponents \( a^{m}\times a^{n}=a^{m + n} \), here \( a = x \), \( m = 2 \), \( n = 1 \), so \( x^{2}\times x=x^{2 + 1}=x^{3} \), and then multiply by 3). For the second term, \( 1\times(3x)=3x \). So, combining these two terms, we have:
\( (f \cdot g)(x)=3x^{3}+3x \)

Answer:

A. \( 3x^{3}+3x \)