1. **Problem:** Given functions $f(x) = 3x - 5$ and $g(x) = 7x + 2$, find each combined function and state any domain restrictions. 2. **Formula and rules:** - Addition: $(f+g)(x) = f(x) + g(x)$ - Subtraction: $(f-g)(x) = f(x) - g(x)$ - Multiplication: $(f \cdot g)(x) = f(x) \times g(x)$ - Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, with domain restriction $g(x) \neq 0$ 3. **Step a: Addition** $$(f+g)(x) = (3x - 5) + (7x + 2)$$ Combine like terms: $$= 3x + 7x - 5 + 2 = 10x - 3$$ No domain restrictions since polynomials are defined for all real $x$. 4. **Step b: Subtraction** $$(f-g)(x) = (3x - 5) - (7x + 2)$$ Distribute the minus: $$= 3x - 5 - 7x - 2 = (3x - 7x) + (-5 - 2) = -4x - 7$$ No domain restrictions. 5. **Step c: Multiplication** $$(f \cdot g)(x) = (3x - 5)(7x + 2)$$ Use FOIL: $$= 3x \times 7x + 3x \times 2 - 5 \times 7x - 5 \times 2$$ $$= 21x^2 + 6x - 35x - 10$$ Combine like terms: $$= 21x^2 - 29x - 10$$ No domain restrictions. 6. **Step d: Division** $$\left(\frac{f}{g}\right)(x) = \frac{3x - 5}{7x + 2}$$ Domain restriction: denominator $\neq 0$ $$7x + 2 \neq 0 \implies 7x \neq -2 \implies x \neq -\frac{2}{7}$$ **Final answers:** - $(f+g)(x) = 10x - 3$, domain all real numbers - $(f-g)(x) = -4x - 7$, domain all real numbers - $(f \cdot g)(x) = 21x^2 - 29x - 10$, domain all real numbers - $\left(\frac{f}{g}\right)(x) = \frac{3x - 5}{7x + 2}$, domain $x \neq -\frac{2}{7}$