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

We are given:   f(x) = x + 1   and   g(x) = 5x² Step 1. Find the functions. 1. f + g:   (f + g)(x) = f(x) + g(x) = (x + 1) + (5x²) = 5x² + x + 1 2. f – g:   (f – g)(x) = f(x) – g(x) = (x + 1) – (5x²) = –5x² + x + 1 3. f · g:   (f · g)(x) = f(x) · g(x) = (x + 1)(5x²) = 5x³ + 5x² 4. f / g:   (f / g)(x) = f(x) / g(x) = (x + 1) / (5x²)   Notice that division by zero is not allowed, so we must have 5x² ≠ 0, i.e. x ≠ 0. Step 2. Determine domains. For polynomial functions (f, g, f+g, f–g, and f·g), the domain is all real numbers, ℝ. a. Domain of f + g:   Since f + g is a polynomial, its domain is all real numbers.   In interval notation, that is (–∞, ∞). b. Domain of f – g:   Also all real numbers: (–∞, ∞). c. Domain of f · g:   Also all real numbers: (–∞, ∞). d. Domain of f / g:   The fraction is defined for all x except where g(x) = 0. Since g(x) = 5x² = 0 when x = 0, the domain is x ∈ ℝ except 0.   In interval notation: (–∞, 0) ∪ (0, ∞). Answering the specific question: What is the domain of f + g? Since (f + g)(x) = 5x² + x + 1 is a polynomial, its domain is (–∞, ∞). Thus, the answer is: B. The domain of f+g is (–∞, ∞).