Question

Use the following functions to complete the table below. f(x) = x^2

g(x) = x h(x) = x^2 -2 *
Even
Odd
Neither
f(x). g(x)
g(x).h(x)
f(x) + g(x)
f(x) + h(x)

Answer 1

To determine if the product or sum of functions is even, odd, or neither, we apply the definitions of even and odd functions. The products and sums of the specific functions f(x) = x^2 and g(x) = x result in one odd function, one even function, and two that are neither even nor odd.

Step-by-step explanation:

To determine whether the products or sums of the given functions are even, odd, or neither, we need to use the properties of even and odd functions. A function f(x) is even if f(x) = f(-x) for all x in the domain of f, which means its graph is symmetric about the y-axis. Conversely, a function f(x) is odd if -f(x) = f(-x), which implies that its graph is symmetric with respect to the origin.

Given the functions f(x) = x², g(x) = x, and h(x) = x² - 2:

• f(x) • g(x) would be x³, which is odd because an even function times an odd function results in an odd function.
• g(x) • h(x) would be x(x² - 2) = x³ - 2x, which is odd.
• f(x) + g(x) would be x² + x, which is neither even nor odd because it does not satisfy the symmetry requirements for either.
• f(x) + h(x) would be x² + (x² - 2) = 2x² - 2, which is even because it is symmetric about the y-axis and satisfies the condition f(x) = f(-x).