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

Let's analyze the functions \( f(x) = 5x + 2 \) and \( g(x) = x - 6 \) by performing the operations \( f + g \), \( f - g \), \( f \cdot g \), and \( \frac{f}{g} \). We'll also determine the domain for each resulting function. --- ### 1. **Addition: \( f + g \)** \[ (f + g)(x) = f(x) + g(x) = (5x + 2) + (x - 6) = 6x - 4 \] **Domain:** Both \( f(x) \) and \( g(x) \) are linear functions defined for all real numbers. \[ \text{Domain of } f + g: (-\infty, \infty) \] --- ### 2. **Subtraction: \( f - g \)** \[ (f - g)(x) = f(x) - g(x) = (5x + 2) - (x - 6) = 4x + 8 \] **Domain:** Both \( f(x) \) and \( g(x) \) are defined for all real numbers. \[ \text{Domain of } f - g: (-\infty, \infty) \] **Simplified Expression:** \[ (f - g)(x) = 4x + 8 \] **Multiple-Choice Question:** *What is the domain of \( f - g \)?* - **A.** The domain of \( f - g \) is \( \varnothing \). - **C.** The domain of \( f - g \) is \( (-\infty, \infty) \). **Correct Answer:** **C.** The domain of \( f - g \) is \( (-\infty, \infty) \). --- ### 3. **Multiplication: \( f \cdot g \)** \[ (f \cdot g)(x) = f(x) \cdot g(x) = (5x + 2)(x - 6) = 5x^2 - 30x + 2x - 12 = 5x^2 - 28x - 12 \] **Domain:** Both \( f(x) \) and \( g(x) \) are polynomials, which are defined for all real numbers. \[ \text{Domain of } f \cdot g: (-\infty, \infty) \] --- ### 4. **Division: \( \frac{f}{g} \)** \[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{5x + 2}{x - 6} \] **Domain:** The denominator \( g(x) = x - 6 \) must not be zero. \[ x - 6 \neq 0 \implies x \neq 6 \] \[ \text{Domain of } \frac{f}{g}: (-\infty, 6) \cup (6, \infty) \] --- ### **Summary of Domains** | Operation | Expression | Domain | |-----------|---------------------|----------------------------| | \( f + g \) | \( 6x - 4 \) | \( (-\infty, \infty) \) | | \( f - g \) | \( 4x + 8 \) | \( (-\infty, \infty) \) | | \( f \cdot g \)| \( 5x^2 - 28x - 12 \) | \( (-\infty, \infty) \) | | \( \frac{f}{g} \)| \( \frac{5x + 2}{x - 6} \)| \( (-\infty, 6) \cup (6, \infty) \) | --- **Conclusion:** For the specific question regarding the domain of \( f - g \), the correct answer is: **C.** The domain of \( f - g \) is \( (-\infty, \infty) \).