1. **Stating the problem:** We have two functions $f(x) = x^2$ and $g(x) = 2x + 1$. We need to find the following combined functions: - $(f+g)(x)$ - $(f-g)(x)$ - $(fg)(x)$ - $\frac{g}{f}(x)$ 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: $\frac{g}{f}(x) = \frac{g(x)}{f(x)}$, provided $f(x) \neq 0$ 3. **Calculations:** - $(f+g)(x) = x^2 + (2x + 1) = x^2 + 2x + 1$ - $(f-g)(x) = x^2 - (2x + 1) = x^2 - 2x - 1$ - $(fg)(x) = x^2 \cdot (2x + 1) = 2x^3 + x^2$ - $\frac{g}{f}(x) = \frac{2x + 1}{x^2}$, with $x \neq 0$ 4. **Explanation:** We simply apply the definitions of sum, difference, product, and quotient of functions by substituting the given expressions and simplifying. **Final answers:** $$(f+g)(x) = x^2 + 2x + 1$$ $$(f-g)(x) = x^2 - 2x - 1$$ $$(fg)(x) = 2x^3 + x^2$$ $$\frac{g}{f}(x) = \frac{2x + 1}{x^2}, \quad x \neq 0$$