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: - $(f + g)(x)$ - $(f - g)(x)$ - $(fg)(x)$ - $\frac{f}{g}(x)$ 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: $$\frac{f}{g}(x) = \frac{x^2}{2x + 1}$$ We must note that $g(x) = 2x + 1 \neq 0$, so $x \neq -\frac{1}{2}$ to avoid division by zero. 4. **Summary:** - $(f + g)(x) = x^2 + 2x + 1$ - $(f - g)(x) = x^2 - 2x - 1$ - $(fg)(x) = 2x^3 + x^2$ - $\frac{f}{g}(x) = \frac{x^2}{2x + 1}$ with $x \neq -\frac{1}{2}$