Odd functions are characterized by the property that \(-\htmlClass{sdt-0000000120}{f(x)} = \htmlClass{sdt-0000000096}{f}(-\htmlClass{sdt-0000000003}{x})\) for all \(\htmlClass{sdt-0000000003}{x}\) in their domain, meaning that if you input the negative of any number into the function, the output will be the negative of what you would get by inputting the number itself. This property results in graphs of odd functions being symmetric with respect to the origin, which means if the graph passes through a point \((\htmlClass{sdt-0000000003}{x},\htmlClass{sdt-0000000017}{y})\), it will also pass through the point \((-\htmlClass{sdt-0000000003}{x},-\htmlClass{sdt-0000000017}{y})\). Examples of odd functions include the sine function \(\htmlClass{sdt-0000000127}{\sin}\) and the cubic function \(\htmlClass{sdt-0000000003}{x}^{3}\). The symmetry of odd functions about the origin makes them particularly useful in modeling situations where a negative input leads to a directly proportional negative output. Understanding odd functions is essential in fields like engineering and physics, where they often describe phenomena like alternating current waveforms or motion patterns.
| \( f(x) \) | The symbol represents a function named \(f\) applied to an input \(\htmlClass{sdt-0000000003}{x}\), producing an output based on a specific rule or relationship. |
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( f \) | This is the symbol for a function. It is commonly used in algebra, and (multivariate) calculus. |
Lets take \(\htmlClass{sdt-0000000120}{f(x)} = \htmlClass{sdt-0000000003}{x}^{3}\) plotted in the figure below. As can be clearly seen the function is symmetrical with respect to the origin(mirrored around the \( \htmlClass{sdt-0000000003}{x} \)-axis and the \( \htmlClass{sdt-0000000017}{y} \)-axis).
Numerically, if we take \(\htmlClass{sdt-0000000003}{x}= 4\), \(-\htmlClass{sdt-0000000120}{f(x)} = -64\) and \(\htmlClass{sdt-0000000096}{f}(-\htmlClass{sdt-0000000003}{x}) = -64\) as well.
