\( (f+g)(x)=5 x^{2}+x+1 \) (Simplify your answer.) What is the domain of \( f+g \) ? A. The domain of \( f+g \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) B. The domain of \( f+g \) is \( \{\quad\} \). (Use a comma to separate answers as needed.) C. The domain of \( f+g \) is \( \varnothing \). \( (f-g)(x)=\square \) (Simplify your answer.)

To solve the problem, we will follow these steps: 1. **Identify the expression for \( (f+g)(x) \)** and simplify it. 2. **Determine the domain of \( f+g \)**. 3. **Find the expression for \( (f-g)(x) \)**. ### Step 1: Simplifying \( (f+g)(x) \) The expression given is: \[ (f+g)(x) = 5x^2 + x + 1 \] This expression is already in its simplest form. ### Step 2: Determining the Domain of \( f+g \) The function \( (f+g)(x) = 5x^2 + x + 1 \) is a polynomial function. The domain of any polynomial function is all real numbers. Therefore, the domain of \( f+g \) is: \[ (-\infty, \infty) \] ### Step 3: Finding \( (f-g)(x) \) To find \( (f-g)(x) \), we need to express it in terms of \( f \) and \( g \). However, we do not have explicit expressions for \( f(x) \) and \( g(x) \). Assuming \( f(x) \) and \( g(x) \) are such that: \[ f(x) + g(x) = 5x^2 + x + 1 \] We can express \( (f-g)(x) \) in terms of \( (f+g)(x) \) and \( (f-g)(x) \) as follows: \[ (f-g)(x) = (f+g)(x) - 2g(x) \] However, without specific values or forms for \( f(x) \) and \( g(x) \), we cannot simplify \( (f-g)(x) \) further. ### Summary of Answers - The domain of \( f+g \) is \( (-\infty, \infty) \). - The expression for \( (f-g)(x) \) cannot be determined without additional information about \( f(x) \) and \( g(x) \). Thus, the answers are: - **Domain of \( f+g \)**: A. The domain of \( f+g \) is \( (-\infty, \infty) \). - **Expression for \( (f-g)(x) \)**: Cannot be determined without more information.