If a function has a symmetry, it is either even or odd. The symmetry is even when the graph is symmetric with respect to the $y$-axis, and odd when it's symmetric about the origin. For example, the function $f(x)=\frac{1}{4}{x}^2$ has even symmetry and the function $g(x)=x^3$ has odd symmetry. If a function has even symmetry, the following rule applies: The rule comes from the fact that even symmetry is a reflection across the $y$-axis. Therefore, changing the sign of the $x$-value does not affect the function value. The concept applies both ways. Hence, if the rule is true for the entire domain, the function has even symmetry. Instead, if a function has odd symmetry, the rule it must follow is An odd symmetry means graphically that the graph is rotated $180^{\circ }$ about the origin. Therefore, changing the sign of the $x$-value also changes the sign of the function value.

If this rule is satisfied on the entire domain, the function has odd symmetry.