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 \( \varnothing \). \( (f-g)(x)=4 x+8 \) (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.) B. The domain of \( f-g \) is . The dompe your answer in interval hof \( f-g \) is \( \varnothing \).

Let's work through each part of the problem step by step. ### Given Functions: - \( f(x) = 5x + 2 \) - \( g(x) = x - 6 \) ### Operations on \( f \) and \( g \): 1. **Sum (\( f + g \))**: \[ (f + g)(x) = f(x) + g(x) = (5x + 2) + (x - 6) = 6x - 4 \] **Domain**: Both \( f \) and \( g \) are linear functions defined for all real numbers. \[ \text{Domain of } (f + g) = \mathbb{R} \quad \text{or} \quad (-\infty, \infty) \] 2. **Difference (\( f - g \))**: \[ (f - g)(x) = f(x) - g(x) = (5x + 2) - (x - 6) = 4x + 8 \] **Domain**: Both \( f \) and \( g \) are defined for all real numbers. \[ \text{Domain of } (f - g) = \mathbb{R} \quad \text{or} \quad (-\infty, \infty) \] 3. **Product (\( f \cdot g \))**: \[ (f \cdot g)(x) = f(x) \times g(x) = (5x + 2)(x - 6) = 5x^2 - 30x + 2x - 12 = 5x^2 - 28x - 12 \] **Domain**: Both \( f \) and \( g \) are defined for all real numbers. \[ \text{Domain of } (f \cdot g) = \mathbb{R} \quad \text{or} \quad (-\infty, \infty) \] 4. **Quotient (\( \frac{f}{g} \))**: \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{5x + 2}{x - 6} \] **Domain**: The denominator \( g(x) = x - 6 \) cannot be zero. \[ x - 6 \neq 0 \quad \Rightarrow \quad x \neq 6 \] \[ \text{Domain of } \left(\frac{f}{g}\right) = \mathbb{R} \setminus \{6\} \quad \text{or} \quad (-\infty, 6) \cup (6, \infty) \] ### Addressing Your Specific Questions: #### Part c: Determining the Domain of \( f + g \) You mentioned: > c. The domain of \( f + g \) is \( \varnothing \). **Correction**: The domain of \( f + g \) is **all real numbers**, not the empty set \( \varnothing \). #### Determining the Domain of \( f - g \): Given: \[ (f - g)(x) = 4x + 8 \] This is a linear function, which is defined for all real numbers. **A. Domain as a Set**: \[ \text{Domain of } (f - g) = \mathbb{R} \] If you need to list specific elements (though unnecessary here), it would include all real numbers. Since it's all-encompassing, no specific values need to be listed. **B. Domain in Interval Notation**: \[ \text{Domain of } (f - g) = (-\infty, \infty) \] #### Summary of Domains: | Function | Domain | |-------------------|-------------------------------| | \( f + g \) | \( \mathbb{R} \) or \( (-\infty, \infty) \) | | \( f - g \) | \( \mathbb{R} \) or \( (-\infty, \infty) \) | | \( f \cdot g \) | \( \mathbb{R} \) or \( (-\infty, \infty) \) | | \( \frac{f}{g} \) | \( \mathbb{R} \setminus \{6\} \) or \( (-\infty, 6) \cup (6, \infty) \) | ### Final Answers for the Domain of \( f - g \): - **A.** The domain of \( f - g \) is \( \mathbb{R} \) (all real numbers). - **B.** The domain of \( f - g \) in interval notation is \( (-\infty, \infty) \).