Question

For each function below, enter the letter e if the function is even, the letter o if it's odd, and the letter n if it's neither even nor odd.

1) e
2) o
3) n

Answer 1

Even and odd functions are classifications based on whether a function has symmetry with respect to the y-axis (even) or the origin (odd). The product of two even functions or two odd functions is always even, but the product of an even and an odd function is neither even nor odd.

Step-by-step explanation:

In mathematics, the concept of even and odd functions is an important part of function analysis. A function f(x) is even if for all x in the function's domain, f(x) = f(-x); geometrically, this means the graph of the function is symmetric with respect to the y-axis. On the other hand, a function is odd if for all x in the function's domain, -f(x) = f(-x), indicating symmetry about the origin. Functions that are neither symmetric about the y-axis nor the origin are neither even nor odd.

For example, the function x2 is even because (-x)2 = x2. A function like x3 is odd because -x3 = (-x)3. When we say an even function times an even function produces an even function, we apply this rule: if f(x) and g(x) are both even functions, then f(x)g(x) will also be an even function because f(x)g(x) = f(-x)g(-x).

In the case of multiplying an odd function by an even function, the resulting function is not guaranteed to be even or odd, so it's typically neither even nor odd. A product of two odd functions, such as x sin(x), results in an even function, adhering to the rule that -f(x)g(x) = f(-x)g(-x), which gives us f(x)g(x), showing that the symmetry about the origin cancels out.