# How do you determine if xy=1 is an even or odd function?

Apr 24, 2016

I'm assuming that $y$ is a function of $x$, so $y \left(x\right) = \frac{1}{x}$. This function is odd.

#### Explanation:

The easiest way to determine whether a function is even or odd is to evaluate $y \left(- x\right)$ in terms of $y \left(x\right)$, e.g.
$y \left(- x\right) = \frac{1}{- x} = - \left(\frac{1}{x}\right) = - y \left(x\right)$
and from the condition $y \left(- x\right) = y \left(x\right)$ we see that our function is odd.

To fully understand this problem let's look at an example of an even function: $y \left(x\right) = {x}^{2}$.
$y \left(- x\right) = {\left(- x\right)}^{2} = {x}^{2} = y \left(x\right)$
Here, since $y \left(- x\right) = y \left(x\right)$ this function is even.

Important notice: some functions, unlike integers, can be both odd and even (e.g. $y \left(x\right) = 0$) or neither odd nor even (e.g. $y \left(x\right) = x + 1$).