Even functions are mathematical functions that exhibit symmetry about the y-axis, characterized by the identity \(\htmlClass{sdt-0000000120}{f(x)} = \htmlClass{sdt-0000000096}{f}(-\htmlClass{sdt-0000000003}{x})\) for all \(\htmlClass{sdt-0000000003}{x}\) in the domain of the function. This means that for any value of \(\htmlClass{sdt-0000000003}{x}\), the function's value at \(\htmlClass{sdt-0000000003}{x}\) is the same as its value at \(-\htmlClass{sdt-0000000003}{x}\), resulting in a mirror image on either side of the \( \htmlClass{sdt-0000000017}{y} \)-axis. Graphically, even functions produce shapes like circles, ellipses, or parabolas that are symmetrical. Common examples of even functions include \(\htmlClass{sdt-0000000003}{x}^{2}\), \(\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000003}{x})\), and \(|\htmlClass{sdt-0000000003}{x}|\), which all demonstrate this kind of symmetry. Understanding even functions is crucial in various fields of mathematics and physics, as they often represent phenomena with inherent symmetries.
| \( f(x) \) | 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. |
This is true by definition
Lets take \(\htmlClass{sdt-0000000120}{f(x)} = \htmlClass{sdt-0000000003}{x}^{2})\) plotted in the figure below. As can be clearly seen the function is symmetrical with respect to the \( \htmlClass{sdt-0000000017}{y} \)-axis(mirrored around the \( \htmlClass{sdt-0000000017}{y} \)-axis).
Numerically, if we take \(\htmlClass{sdt-0000000003}{x} = 4\), \(\htmlClass{sdt-0000000120}{f(x)} = 16\) and \(\htmlClass{sdt-0000000096}{f}(-\htmlClass{sdt-0000000003}{x}) = 16\) as well.
