Questions: Find each of the following functions and state their domains. (Enter the domains in interval notation.) f(x)=x^3+3 x^2, g(x)=7 x^2-1 (a) f+g f+g= domain

Transcript text: Find each of the following functions and state their domains. (Enter the domains in interval notation.) \[ f(x)=x^{3}+3 x^{2}, \quad g(x)=7 x^{2}-1 \] (a) $f+g$ \[ f+g= \] domain $\square$

Solution

Solution Steps

To find $ f+g $, we need to add the functions $ f(x) $ and $ g(x) $. The domain of $ f+g $ will be the intersection of the domains of $ f $ and $ g $. Since both functions are polynomials, their domains are all real numbers.

Step 1: Define the Functions

We are given two functions: \[ f(x) = x^3 + 3x^2 \] \[ g(x) = 7x^2 - 1 \]

Step 2: Add the Functions

To find $ f+g $, we add the two functions: \[ f(x) + g(x) = (x^3 + 3x^2) + (7x^2 - 1) \] \[ f(x) + g(x) = x^3 + 3x^2 + 7x^2 - 1 \] \[ f(x) + g(x) = x^3 + 10x^2 - 1 \]

Step 3: Determine the Domain

The domain of a polynomial function is all real numbers. Since both $ f(x) $ and $ g(x) $ are polynomials, their domains are all real numbers. Therefore, the domain of $ f+g $ is also all real numbers.