Find \( f+g, f-g \), fg and \( \frac{f}{g} \). Determine the domain for each function. \[ f(x)=5 x+8, g(x)=x+6 \] \( (\mathrm{f}-\mathrm{g})(\mathrm{x})=4 \mathrm{x}+2 \) (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 neede C. The domain of \( f-g \) is \( \varnothing \). \[ (f g)(x)=5 x^{2}+38 x+48 \] (Simplify your answer.) What is the domain of fg ? A. The domain of \( f g \) is \( \square \) \} (Use a comma to separate answers as needed ) B. The domain of \( f g \) is \( \square \) (Type your answer in interval notation.) C. The domain of \( f g \) is \( \varnothing \)

Let's analyze the functions \( f(x) = 5x + 8 \) and \( g(x) = x + 6 \), and determine their sum, difference, product, and quotient along with their domains. ### 1. Sum: \( f + g \) \[ (f + g)(x) = (5x + 8) + (x + 6) = 6x + 14 \] **Domain:** Both \( f(x) \) and \( g(x) \) are defined for all real numbers. Therefore, the domain of \( f + g \) is: \[ (-\infty, \infty) \] ### 2. Difference: \( f - g \) \[ (f - g)(x) = (5x + 8) - (x + 6) = 4x + 2 \] **Domain:** Both \( f(x) \) and \( g(x) \) are defined for all real numbers. Therefore, the domain of \( f - g \) is: \[ (-\infty, \infty) \] **Correct Option:** - **A.** The domain of \( f - g \) is \( (-\infty, \infty) \). ### 3. Product: \( fg \) \[ (fg)(x) = (5x + 8)(x + 6) = 5x^2 + 38x + 48 \] **Domain:** Both \( f(x) \) and \( g(x) \) are defined for all real numbers. Therefore, the domain of \( fg \) is: \[ (-\infty, \infty) \] **Correct Option:** - **B.** The domain of \( fg \) is \( (-\infty, \infty) \). ### 4. Quotient: \( \frac{f}{g} \) \[ \left(\frac{f}{g}\right)(x) = \frac{5x + 8}{x + 6} \] **Domain:** The function \( g(x) \) cannot be zero to avoid division by zero. Therefore, \( x + 6 \neq 0 \) which implies \( x \neq -6 \). Thus, the domain is: \[ (-\infty, -6) \cup (-6, \infty) \] ### Summary of Domains: - **\( f + g \):** \( (-\infty, \infty) \) - **\( f - g \):** \( (-\infty, \infty) \) **[Option A]** - **\( fg \):** \( (-\infty, \infty) \) **[Option B]** - **\( \frac{f}{g} \):** \( (-\infty, -6) \cup (-6, \infty) \)