# Even and Odd Functions

Even and odd functions are functions that satisfy certain properties. This is a powerful concept; identifying even and odd functions can make some seemingly tough integration problems trivial.

**Even functions** are functions that satisfy $f(x) = f(-x)$ for all $x$. Even functions are symmetric about the line $x =0$.

**Odd functions** are functions that satisfy $f(x) = -f(-x)$ for all $x$. Odd functions exhibit point symmetry about the origin.

It is possible for a function to be neither odd nor even.

The function $f(x) = \displaystyle \dfrac{x}{1-2^{x}}-\dfrac{x}{2}$ is $\text{\_\_\_\_\_\_\_\_\_\_}.$

$\quad \text{A) }$ even but not odd

$\quad \text{B) }$ odd but not even

$\quad \text{C) }$ both even and odd

$\quad \text{D) }$ neither even nor odd

If we evaluate $f(-x)$, we get:

$\begin{aligned} f(-x)& = \dfrac{-x}{1-2^{-x}} - \dfrac{-x}{2} \\ &= \dfrac{-x \cdot 2^x}{2^x -1} + \dfrac{x}{2} \\ &= \dfrac{x}{1-2^x} - x + \dfrac{x}{2} \\ &= \dfrac{x}{1-2^x} - \dfrac{x}{2} \\ &= f(x). \end{aligned}$

Therefore, $f(x)$ is even and obviously not odd, so the answer is choice $\text{A)}$. $_\square$

Even and odd functions can also be spotted by their graphs - specifically, even functions have the y-axis as a line of symmetry and odd functions have rotational symmetry about the origin. For example, take a point (a,b) on the coordinate plane such that the function f(x) is even, thus f(x) = f(-x) - so if (a,b) is on the graph, (-a,b) is as well. Odd functions can be rotated 180 degrees about the origin and then appear the same, and so are said to be symmetric about the origin.

**Cite as:**Even and Odd Functions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/even-and-odd-functions/