Question: Functions
Sum, difference, and product of two functions
Suppose that the functions $r$ and $s$ are defined for all real numbers $x$ as follows.
$r(x) = 2x^2$
$s(x) = x + 3$
Find the following.
$(s + r)(x) =$
$(s - r)(x) =$
$(s \cdot r)(-1) =$
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. **Recall the formulas:**
- Sum of functions: $(s + r)(x) = s(x) + r(x)$
- Difference of functions: $(s - r)(x) = s(x) - r(x)$
- Product of functions: $(s \cdot r)(x) = s(x) \times r(x)$
3. **Calculate the sum:**
$$(s + r)(x) = s(x) + r(x) = (x + 3) + 2x^2 = 2x^2 + x + 3$$
4. **Calculate the difference:**
$$(s - r)(x) = s(x) - r(x) = (x + 3) - 2x^2 = -2x^2 + x + 3$$
5. **Calculate the product at $x = -1$:**
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:**
- $(s + r)(x) = 2x^2 + x + 3$
- $(s - r)(x) = -2x^2 + x + 3$
- $(s \cdot r)(-1) = 4$