Find \( f+g, f-g \), fg and \( \frac{f}{g} \). Determine the domain for each function. \( f(x)=5 x+8, g(x)=x+6 \) \( (f+g)(x)=\square \) (Simplify your answer.)

Given the functions \( f(x) = 5x + 8 \) and \( g(x) = x + 6 \), let's find \( f + g \), \( f - g \), \( fg \), and \( \frac{f}{g} \), along with their respective domains. ### 1. Sum: \( (f + g)(x) \) \[ \begin{align*} (f + g)(x) &= f(x) + g(x) \\ &= (5x + 8) + (x + 6) \\ &= 5x + x + 8 + 6 \\ &= 6x + 14 \end{align*} \] **Domain:** All real numbers, \( \mathbb{R} \). --- ### 2. Difference: \( (f - g)(x) \) \[ \begin{align*} (f - g)(x) &= f(x) - g(x) \\ &= (5x + 8) - (x + 6) \\ &= 5x - x + 8 - 6 \\ &= 4x + 2 \end{align*} \] **Domain:** All real numbers, \( \mathbb{R} \). --- ### 3. Product: \( (fg)(x) \) \[ \begin{align*} (fg)(x) &= f(x) \cdot g(x) \\ &= (5x + 8)(x + 6) \\ &= 5x \cdot x + 5x \cdot 6 + 8 \cdot x + 8 \cdot 6 \\ &= 5x^2 + 30x + 8x + 48 \\ &= 5x^2 + 38x + 48 \end{align*} \] **Domain:** All real numbers, \( \mathbb{R} \). --- ### 4. Quotient: \( \left(\frac{f}{g}\right)(x) \) \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{5x + 8}{x + 6} \] **Domain:** All real numbers except \( x = -6 \), since the denominator cannot be zero. \[ \text{Domain}: \quad x \in \mathbb{R} \setminus \{-6\} \] --- ### Summary - **Sum:** \( (f + g)(x) = 6x + 14 \)  **Domain:** \( \mathbb{R} \) - **Difference:** \( (f - g)(x) = 4x + 2 \)  **Domain:** \( \mathbb{R} \) - **Product:** \( (fg)(x) = 5x^2 + 38x + 48 \)  **Domain:** \( \mathbb{R} \) - **Quotient:** \( \frac{f}{g}(x) = \frac{5x + 8}{x + 6} \)  **Domain:** \( x \in \mathbb{R}, x \neq -6 \)