Odd and even functions (HL)

While occasional questions could specifically ask about $f(-x)$, many problems are intended to be simplified by applying properties of odd and even functions. Knowing about these types of functions can also enhance understanding of trigonometric functions.

Contents

• Spotting an odd or even function in the wild
• Properties

Spotting an odd or even function in the wild

Odd functions are rotationally symmetric about $(0, 0)$, meaning that if the entire graph is rotated $180\degree$, it coincides on itself. Algebraically

Common odd functions include $x \mapsto \sin x$, $x \mapsto \tan x$, $x \mapsto x$, $x \mapsto x^3$, $x \mapsto \frac 1x$.

Even functions are symmetric about $x = 0$, meaning that the left side of the graph can be reflected and coincided on top of the right side. Algebraically

Common even functions include $x \mapsto \cos x$, $x \mapsto x^2$, $x \mapsto \frac{1}{x^2}$, $x \mapsto \lvert x\rvert$

Usually, horizontal translations break the symmetry. Even functions can have vertical translations while maintaining the symmetry. Odd and even functions do not have to pass through $(0, 0)$.

Properties

Linear combinations (sums of multiples), their inverses (if exists) of odd (even) functions are also odd (even).

For some odd $f$ and $g$ and even $h$ and $j$, their composite functions are even if one of the function is even and odd when both are odd.

This implies reciprocals of odd (even) functions are also odd (even). Furthermore, the composition of any function with an even function, is even.

All features are also symmetric for odd and even functions. A notable exception is that maxima on one side of an odd function become minima on the other side.

In the Maclaurin series expansions, odd functions only have odd powers of $x$, while even functions only have even powers.

Additional properties:

These assume the function is differentiable or integrable, where applicable.

For additional properties, see the Wikipedia article on even and odd functions.