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: $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\frac{f}{g}(x)$. 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: $\frac{f}{g}(x) = \frac{f(x)}{g(x)}$, with $g(x) \neq 0$ 3. **Calculate each:** - $(f+g)(x) = x^2 + (2x + 1) = x^2 + 2x + 1$ - $(f-g)(x) = x^2 - (2x + 1) = x^2 - 2x - 1$ - $(fg)(x) = x^2 \cdot (2x + 1) = 2x^3 + x^2$ - $\frac{f}{g}(x) = \frac{x^2}{2x + 1}$, where $2x + 1 \neq 0$ (i.e., $x \neq -\frac{1}{2}$) 4. **Final answers:** - $(f+g)(x) = x^2 + 2x + 1$ - $(f-g)(x) = x^2 - 2x - 1$ - $(fg)(x) = 2x^3 + x^2$ - $\frac{f}{g}(x) = \frac{x^2}{2x + 1}$ with $x \neq -\frac{1}{2}$