# 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:

\[\begin{align} f(-x)& = \dfrac{-x}{1-2^{-x}} - \dfrac{-x}{2} \\ &= \dfrac{-x \cdot 2^x}{2^x -1} + \dfrac{x}{2} \\ &= \dfrac{x}{1-2^x} - x + \dfrac{x}{2} \\ &= \dfrac{x}{1-2^x} - \dfrac{x}{2} \\ &= f(x). \end{align} \]

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/