Function Operations 028E39

1. **State the problem:** We are given two functions $r(x) = 2x^2$ and $s(x) = x + 3$. We need to find the sum $(s + r)(x)$, the difference $(s - r)(x)$, and the product $(s \cdot r)(-1)$. 2. **Sum of functions:** The sum $(s + r)(x)$ is defined as $s(x) + r(x)$. $$ (s + r)(x) = s(x) + r(x) = (x + 3) + 2x^2 = 2x^2 + x + 3 $$ 3. **Difference of functions:** The difference $(s - r)(x)$ is defined as $s(x) - r(x)$. $$ (s - r)(x) = s(x) - r(x) = (x + 3) - 2x^2 = -2x^2 + x + 3 $$ Note: The original incorrect answer showed $(s - r)(x) = 2x^2 + x + 3$, which is wrong because the subtraction changes the sign of $2x^2$. 4. **Product of functions evaluated at $x = -1$:** The product $(s \cdot r)(x)$ is defined as $s(x) \times r(x)$. First, find $s(-1)$ and $r(-1)$: $$ s(-1) = (-1) + 3 = 2 $$ $$ r(-1) = 2(-1)^2 = 2(1) = 2 $$ Then multiply: $$ (s \cdot r)(-1) = s(-1) \times r(-1) = 2 \times 2 = 4 $$ **Final answers:** - $(s + r)(x) = 2x^2 + x + 3$ - $(s - r)(x) = -2x^2 + x + 3$ - $(s \cdot r)(-1) = 4$