1. **Problem statement:** Given two functions $f(x) = x^2$ and $g(x) = 2x + 1$, find the following combined functions: $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\frac{f}{g}(x)$. 2. **Formulas and rules:** - Sum of functions: $(f+g)(x) = f(x) + g(x)$ - Difference of functions: $(f-g)(x) = f(x) - g(x)$ - Product of functions: $(fg)(x) = f(x) \cdot g(x)$ - Quotient of functions: $\frac{f}{g}(x) = \frac{f(x)}{g(x)}$, with $g(x) \neq 0$ 3. **Calculate each combined function:** - Sum: $$(f+g)(x) = x^2 + (2x + 1) = x^2 + 2x + 1$$ - Difference: $$(f-g)(x) = x^2 - (2x + 1) = x^2 - 2x - 1$$ - Product: $$(fg)(x) = x^2 \cdot (2x + 1) = 2x^3 + x^2$$ - Quotient: $$\frac{f}{g}(x) = \frac{x^2}{2x + 1}$$ Note: The domain excludes values where $2x + 1 = 0$, 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 $x \neq -\frac{1}{2}$