Final answer:
To determine if functions f and g are odd, even, or neither, we look at symmetries: even functions are symmetric about the y-axis (y(x) = y(-x)), and odd functions are antisymmetric (y(x) = −y(-x)). The product of two even or two odd functions results in an even function, while the product of an odd function with an even one yields an odd function, with integrals over all space being zero for odd functions.
Step-by-step explanation:
To determine whether functions f and g are odd, even, or neither, we need to understand the definitions of even and odd functions. An even function satisfies the condition y(x) = y(-x), which means it is symmetric about the y-axis. On the other hand, an odd function satisfies y(x) = −y(-x), showing antisymmetry, meaning it reflects about the y-axis and then about the x-axis
Product of odd functions: Multiplying two odd functions results in an even function, such as x sin x (odd times odd is even).
Product of an odd and an even function: When an odd function is multiplied by an even function, the result is an odd function. For instance, xe−x² (odd times even is odd).
Additionally, the integral of an odd function over the entire space is zero due to the symmetrical cancellation above and below the x-axis. This property can simplify expectation-value calculations in contexts such as wave functions in physics.