Question: Functions
Sum, difference, and product of two functions
Suppose that the functions $g$ and $h$ are defined for all real numbers $x$ as follows.
g(x) = 4x^3
h(x) = x^2
Find the following.
$(h + g)(x) = \square$
$(h \cdot g)(x) = \square$
$(h - g)(-1) = \square$
1. **State the problem:** We are given two functions $g(x) = 4x^3$ and $h(x) = x^2$ and need to find the sum, product, and difference of these functions at specified inputs.
2. **Sum of functions:** The sum $(h + g)(x)$ is defined as $h(x) + g(x)$.
$$ (h + g)(x) = h(x) + g(x) = x^2 + 4x^3 $$
3. **Product of functions:** The product $(h \cdot g)(x)$ is defined as $h(x) \times g(x)$.
$$ (h \cdot g)(x) = h(x) \times g(x) = x^2 \times 4x^3 = 4x^{2+3} = 4x^5 $$
4. **Difference of functions at $x = -1$:** The difference $(h - g)(-1)$ is defined as $h(-1) - g(-1)$.
Calculate each:
$$ h(-1) = (-1)^2 = 1 $$
$$ g(-1) = 4(-1)^3 = 4(-1) = -4 $$
So,
$$ (h - g)(-1) = 1 - (-4) = 1 + 4 = 5 $$
**Final answers:**
- $(h + g)(x) = x^2 + 4x^3$
- $(h \cdot g)(x) = 4x^5$
- $(h - g)(-1) = 5$