1. **State the problem:** Given two functions $f(x) = x^2$ and $g(x) = 2x + 1$, find the following combined functions: $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $(f/g)(x)$. 2. **Recall the formulas:** - Sum: $(f+g)(x) = f(x) + g(x)$ - Difference: $(f-g)(x) = f(x) - g(x)$ - Product: $(fg)(x) = f(x) imes g(x)$ - Quotient: $(f/g)(x) = \frac{f(x)}{g(x)}$, with $g(x) \neq 0$ 3. **Calculate each:** - 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: $$(fg)(x) = x^2 \times (2x + 1) = 2x^3 + x^2$$ - Quotient: $$(f/g)(x) = \frac{x^2}{2x + 1}$$ 4. **Explain domain for quotient:** The function $(f/g)(x)$ is defined for all real $x$ except where the denominator is zero, i.e., where $2x + 1 = 0 \Rightarrow x = -\frac{1}{2}$. **Final answers:** $$(f+g)(x) = x^2 + 2x + 1$$ $$(f-g)(x) = x^2 - 2x - 1$$ $$(fg)(x) = 2x^3 + x^2$$ $$(f/g)(x) = \frac{x^2}{2x + 1}, \quad x \neq -\frac{1}{2}$$