\( 1 \leftarrow \) Find \( f+g, f-g, f g \), and \( \frac{f}{g} \). Determine the domain for each function. \( f(x)=\sqrt{x} ; g(x)=x-9 \) A. The domain of \( f g \) is \( [0, \infty) \). (Type your answer in interval notation.) C. The domain of \( f g \) is \( \varnothing \). \( \left(\frac{f}{g}\right)(x)=\square \) (Simplify your answer.)

Let's work through each part of the problem step by step. **Given Functions:** - \( f(x) = \sqrt{x} \) - \( g(x) = x - 9 \) --- ### 1. **Sum: \( f + g \)** \[ (f + g)(x) = f(x) + g(x) = \sqrt{x} + (x - 9) = x + \sqrt{x} - 9 \] **Domain:** - **\( f(x) = \sqrt{x} \)** is defined for \( x \geq 0 \). - **\( g(x) = x - 9 \)** is defined for all real numbers (\( \mathbb{R} \)). **Intersection of Domains:** \[ \text{Domain of } (f + g) = [0, \infty) \] --- ### 2. **Difference: \( f - g \)** \[ (f - g)(x) = f(x) - g(x) = \sqrt{x} - (x - 9) = -x + \sqrt{x} + 9 \] **Domain:** Same as the sum. \[ \text{Domain of } (f - g) = [0, \infty) \] --- ### 3. **Product: \( f \cdot g \)** \[ (f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x - 9) \] **Domain:** Same as the sum. \[ \text{Domain of } (f \cdot g) = [0, \infty) \] *Note:* Among the options provided: - **A.** The domain of \( f \cdot g \) is \( [0, \infty) \). **(Correct)** - **C.** The domain of \( f \cdot g \) is \( \varnothing \). **(Incorrect)** --- ### 4. **Quotient: \( \frac{f}{g} \)** \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{x - 9} \] **Simplified Form:** \[ \left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x - 9} \] **Domain:** - **\( f(x) = \sqrt{x} \):** \( x \geq 0 \) - **\( g(x) = x - 9 \):** \( x \neq 9 \) (since division by zero is undefined) **Combined Domain:** \[ \text{Domain of } \left(\frac{f}{g}\right) = [0, 9) \cup (9, \infty) \] --- ### **Summary of Results:** 1. **Sum:** \( f + g = x + \sqrt{x} - 9 \) **Domain:** \( [0, \infty) \) 2. **Difference:** \( f - g = -x + \sqrt{x} + 9 \) **Domain:** \( [0, \infty) \) 3. **Product:** \( f \cdot g = \sqrt{x} \cdot (x - 9) \) **Domain:** \( [0, \infty) \) 4. **Quotient:** \( \frac{f}{g} = \frac{\sqrt{x}}{x - 9} \) **Domain:** \( [0, 9) \cup (9, \infty) \) --- **Final Answer for the Quotient Function:** \[ \left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x - 9} \] --- Feel free to ask if you need further clarification on any of these steps!