1. **State the problem:** We are given two functions $f(x) = 4x - 3$ and $g(x) = x - 2$. We need to find the functions $f+g$, $f-g$, $fg$, and $\frac{f}{g}$, and determine the domain of each. 2. **Find $f+g$:** $$(f+g)(x) = f(x) + g(x) = (4x - 3) + (x - 2)$$ Simplify: $$(f+g)(x) = 4x - 3 + x - 2 = 5x - 5$$ 3. **Find $f-g$:** $$(f-g)(x) = f(x) - g(x) = (4x - 3) - (x - 2)$$ Simplify: $$(f-g)(x) = 4x - 3 - x + 2 = 3x - 1$$ 4. **Find $fg$ (product):** $$(fg)(x) = f(x) \cdot g(x) = (4x - 3)(x - 2)$$ Multiply: $$(fg)(x) = 4x \cdot x - 4x \cdot 2 - 3 \cdot x + 3 \cdot 2 = 4x^2 - 8x - 3x + 6$$ Simplify: $$(fg)(x) = 4x^2 - 11x + 6$$ 5. **Find $\frac{f}{g}$ (quotient):** $$(\frac{f}{g})(x) = \frac{f(x)}{g(x)} = \frac{4x - 3}{x - 2}$$ 6. **Determine domains:** - $f(x)$ and $g(x)$ are linear functions, so their domains are all real numbers. - For $f+g$, $f-g$, and $fg$, the domain is all real numbers because sums, differences, and products of polynomials are defined everywhere. - For $\frac{f}{g}$, the denominator $g(x) = x - 2$ cannot be zero, so $x \neq 2$. **Final answers:** $$(f+g)(x) = 5x - 5$$ $$(f-g)(x) = 3x - 1$$ $$(fg)(x) = 4x^2 - 11x + 6$$ $$(\frac{f}{g})(x) = \frac{4x - 3}{x - 2}, \quad x \neq 2$$