1. **State the problem:** We are given two functions: $$f(x) = 5x - 6, \quad g(x) = x + 8$$ We need to find the following combined functions: - $f + g$ - $f - g$ - $fg$ - $\frac{f}{g}$ and determine the domain of each. 2. **Formulas and rules:** - Sum: $(f+g)(x) = f(x) + g(x)$ - Difference: $(f-g)(x) = f(x) - g(x)$ - Product: $(fg)(x) = f(x) \cdot g(x)$ - Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, domain excludes points where $g(x) = 0$ 3. **Calculate each combined function:** - Sum: $$(f+g)(x) = (5x - 6) + (x + 8) = 5x - 6 + x + 8 = 6x + 2$$ - Difference: $$(f-g)(x) = (5x - 6) - (x + 8) = 5x - 6 - x - 8 = 4x - 14$$ - Product: $$(fg)(x) = (5x - 6)(x + 8) = 5x \cdot x + 5x \cdot 8 - 6 \cdot x - 6 \cdot 8 = 5x^2 + 40x - 6x - 48 = 5x^2 + 34x - 48$$ - Quotient: $$\left(\frac{f}{g}\right)(x) = \frac{5x - 6}{x + 8}$$ 4. **Determine domains:** - For $f+g$, $f-g$, and $fg$, both $f$ and $g$ are polynomials defined for all real numbers, so domain is $\mathbb{R}$. - For $\frac{f}{g}$, denominator $g(x) = x + 8 \neq 0$, so domain excludes $x = -8$. 5. **Final answers:** - $$(f+g)(x) = 6x + 2, \quad \text{domain: } (-\infty, \infty)$$ - $$(f-g)(x) = 4x - 14, \quad \text{domain: } (-\infty, \infty)$$ - $$(fg)(x) = 5x^2 + 34x - 48, \quad \text{domain: } (-\infty, \infty)$$ - $$\left(\frac{f}{g}\right)(x) = \frac{5x - 6}{x + 8}, \quad \text{domain: } (-\infty, -8) \cup (-8, \infty)$$