Find \( f+g, f-g \), \( g \) and \( \frac{f}{g} \). Determine the domain for each function. \( f(x)=5 x+8, g(x)=x-3 \) ( \( f+g)(x)=6 x+5 \) (Simplify your answer.) What is the domain of \( f+g \) ? The domain of \( f+g \) is \( \{ \). (Use a comma to separate answers as needed.) A. Thes

**Step 1. Compute \(f+g\)** We have \[ f(x) = 5x + 8 \quad \text{and} \quad g(x) = x - 3. \] Then, \[ (f+g)(x) = f(x) + g(x) = (5x+8) + (x-3) = 6x+5. \] **Step 2. Determine the domain of \(f+g\)** Since both \(f(x)\) and \(g(x)\) are defined for all real numbers, the domain of \(f+g\) is the set of all real numbers: \[ \text{Domain of } (f+g) = (-\infty, \infty). \] **Step 3. Compute \(f-g\)** \[ (f-g)(x) = f(x) - g(x) = (5x+8) - (x-3) = 5x+8-x+3 = 4x+11. \] **Step 4. Determine the domain of \(f-g\)** Again, since \(f(x)\) and \(g(x)\) are defined for all real numbers, \[ \text{Domain of } (f-g) = (-\infty, \infty). \] **Step 5. State \(g\) and its domain** The function \(g\) is given as: \[ g(x)=x-3. \] Its domain is also all real numbers: \[ \text{Domain of } g = (-\infty, \infty). \] **Step 6. Compute \(\frac{f}{g}\)** \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{5x+8}{x-3}. \] **Step 7. Determine the domain of \(\frac{f}{g}\)** The expression \(\frac{5x+8}{x-3}\) is defined for all \(x\) except where the denominator is zero. Set the denominator equal to zero: \[ x-3=0 \quad \Rightarrow \quad x=3. \] Thus, \[ \text{Domain of } \frac{f}{g} = (-\infty, 3)\cup(3, \infty). \] **Final Answers** - \( (f+g)(x)=6x+5 \) with domain \( (-\infty, \infty) \). - \( (f-g)(x)=4x+11 \) with domain \( (-\infty, \infty) \). - \( g(x)=x-3 \) with domain \( (-\infty, \infty) \). - \(\displaystyle \left(\frac{f}{g}\right)(x)=\frac{5x+8}{x-3} \) with domain \( (-\infty, 3)\cup(3, \infty) \).