1. **State the problem:** Given two functions $f(x) = x^2$ and $g(x) = 2x + 1$, find the following combined functions: - $(f + g)(x)$ - $(f - g)(x)$ - $(fg)(x)$ - $\frac{f}{g}(x)$ 2. **Recall the formulas for combined functions:** - 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 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: $$ \frac{f}{g}(x) = \frac{x^2}{2x + 1} $$ 4. **Important note:** For the quotient, the function is defined only where $2x + 1 \neq 0$, i.e., $x \neq -\frac{1}{2}$. **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 $x \neq -\frac{1}{2}$