A relation may exhibit symmetry if it behaves consistently upon substitution of x with -x. Even functions are symmetric about the y-axis and satisfy y(x) = y(-x), while odd functions are symmetric about the origin and satisfy y(x) = -y(-x). Assessing symmetry helps in various problem types including function analysis and quantum mechanical calculations.
To determine if a relation has symmetry, we consider the function's behavior when we replace x with -x. Here is how:
Even functions are symmetric with respect to the y-axis. An even function satisfies the condition y(x) = y(-x). Function notation for an even function is f(x). A line of y-axis symmetry divides the graph into halves that are reflections of each other.
Odd functions are symmetric with respect to the origin. They satisfy the condition y(x) = -y(-x). Odd functions can be rotated 180 degrees about the origin and look the same. Function notation for an odd function is f(x)
Common problem types include determining function symmetry, performing expectation-value calculations in quantum mechanics where symmetric wave functions can simplify the process, and analyzing charge density in spherical coordinates to find spherical symmetry.
As a side note, multiplying even functions together, or odd functions together, typically results in an even function, while an even function multiplied by an odd function gives an odd function.