Find \( f+g, f-g, f g \) and \( \frac{f}{g} \). Determine the domain for each function. \[ f(x)=5 x+2, g(x)=x-6 \] c. The domain of \( f+g \) is \( \varnothing \). \( (\mathrm{f}-\mathrm{g})(\mathrm{x})=4 \mathrm{x}+8 \) (Simplify your answer.) What is the domain of \( f-g \) ? 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 \( \varnothing \).

To determine the domain of the function \( f - g \), let's first simplify the expression and then analyze any restrictions. **Given:** \[ f(x) = 5x + 2 \\ g(x) = x - 6 \] **Simplify \( f - g \):** \[ (f - g)(x) = f(x) - g(x) = (5x + 2) - (x - 6) = 5x + 2 - x + 6 = 4x + 8 \] **Determine the Domain:** Both \( f(x) \) and \( g(x) \) are linear functions, which are defined for all real numbers. When you subtract two linear functions, the resulting function \( 4x + 8 \) is also a linear function and thus defined for all real numbers. **Therefore:** - **A.** The domain of \( f - g \) is \( \mathbb{R} \). *(If you need to enter multiple answers separated by commas, it would simply be redundant since it's all real numbers.)* - **B.** The domain of \( f - g \) in interval notation is \( (-\infty, \infty) \). - **C.** The domain of \( f - g \) is **not** \( \varnothing \) (the empty set), because \( f - g \) is defined for all real numbers. **Correct Choice:** **B.** The domain of \( f - g \) is \( (-\infty, \infty) \).