Find \( f+g, f-g, f g \), and \( \frac{f}{g} \). Determine the domain for each function. \[ f(x)=\sqrt{x} ; g(x)=x-9 \] \( (f+g)(x)=\sqrt{x}+x-9 \) (Simplify your answer.) What is the domain of \( f+g \) ? A. The domain of \( \mathrm{f}+\mathrm{g} \) is \( \{\quad \). (Use a comma to separate answers as needed B. The domain of \( f+g \) is \( [0, \infty) \) (Type your answer in interval notation.) C. The domain of \( f+g \) is \( \varnothing \) \( (f-g)(x)=\sqrt{x}-x+9 \) (Simplify your answer.) What is the domain of \( f-g \) ? A. The domain of \( f-g \) is \( [0, \infty) \). (Type your answer in interval notation.) B. The domain of \( f-g \) is \( \} \) (Use a comma to separate answers as needed C. The domain of \( f-g \) is \( \varnothing \) \( (f g)(x)= \) \( \square \) (Simplify your answer.)

Certainly! Let's analyze the functions \( f(x) = \sqrt{x} \) and \( g(x) = x - 9 \) by finding their sum, difference, product, and quotient, along with determining the domain for each resulting function. --- ### 1. **Sum: \( f + g \)** \[ (f + g)(x) = \sqrt{x} + (x - 9) = \sqrt{x} + x - 9 \] **Domain of \( f + g \):** - **\( f(x) = \sqrt{x} \):** Defined for \( x \geq 0 \). - **\( g(x) = x - 9 \):** Defined for all real numbers \( x \). **Intersection of Domains:** \( x \geq 0 \). **Answer Choices:** - **A.** The domain of \( f + g \) is \( \{\quad \} \). *(Requires specific values; not applicable here.)* - **B.** The domain of \( f + g \) is \( [0, \infty) \). - **C.** The domain of \( f + g \) is \( \varnothing \) (the empty set). **Correct Choice:** **B.** The domain of \( f + g \) is \( [0, \infty) \). --- ### 2. **Difference: \( f - g \)** \[ (f - g)(x) = \sqrt{x} - (x - 9) = \sqrt{x} - x + 9 \] **Domain of \( f - g \):** - **\( f(x) = \sqrt{x} \):** Defined for \( x \geq 0 \). - **\( g(x) = x - 9 \):** Defined for all real numbers \( x \). **Intersection of Domains:** \( x \geq 0 \). **Answer Choices:** - **A.** The domain of \( f - g \) is \( [0, \infty) \). - **B.** The domain of \( f - g \) is \( \{\quad \} \). *(Requires specific values; not applicable here.)* - **C.** The domain of \( f - g \) is \( \varnothing \). **Correct Choice:** **A.** The domain of \( f - g \) is \( [0, \infty) \). --- ### 3. **Product: \( fg \)** \[ (fg)(x) = \sqrt{x} \cdot (x - 9) = (x - 9)\sqrt{x} \] **Domain of \( fg \):** - **\( f(x) = \sqrt{x} \):** Defined for \( x \geq 0 \). - **\( g(x) = x - 9 \):** Defined for all real numbers \( x \). **Intersection of Domains:** \( x \geq 0 \). **Simplified Expression:** \[ (fg)(x) = (x - 9)\sqrt{x} \] **Domain:** \( [0, \infty) \) --- ### 4. **Quotient: \( \frac{f}{g} \)** \[ \left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x - 9} \] **Domain of \( \frac{f}{g} \):** - **\( f(x) = \sqrt{x} \):** Defined for \( x \geq 0 \). - **Denominator \( g(x) = x - 9 \):** Must not be zero, so \( x \neq 9 \). **Intersection of Domains:** \( x \geq 0 \) **and** \( x \neq 9 \). **Domain in Interval Notation:** \[ [0, 9) \cup (9, \infty) \] --- ### **Summary:** - **\( (f + g)(x) = \sqrt{x} + x - 9 \)** - **Domain:** \( [0, \infty) \) - **\( (f - g)(x) = \sqrt{x} - x + 9 \)** - **Domain:** \( [0, \infty) \) - **\( (fg)(x) = (x - 9)\sqrt{x} \)** - **Domain:** \( [0, \infty) \) - **\( \left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x - 9} \)** - **Domain:** \( [0, 9) \cup (9, \infty) \) --- Feel free to ask if you need further clarification on any of these functions or their domains!