# What is the domain of the function?

Apr 8, 2018

$x \in \left(- \infty , \infty\right)$
$y \in \left(\infty , 0\right) \cup \left(0 , \infty\right)$

#### Explanation:

Recall that the domain is the set of valid inputs to the function. In this case, it is the set of all pairs $\left(x , y\right)$ that are valid inputs.

Also recall that, while you cannot take the square root of a negative number, you can take the cube root of a negative number. For example, $\sqrt[3]{- 8} = - 2$. Thus, $\sqrt[3]{x}$ is valid for any value of $x$.

Additionally, $\sqrt[3]{y}$ is valid for all inputs of $y$, but our function $f \left(x , y\right) = \frac{\sqrt[3]{x}}{\sqrt[3]{y}}$ is undefined when $y = 0$, due to division by zero.

Thus, our domain is $x \in \left(- \infty , \infty\right)$, $y \in \left(- \infty , 0\right) \cup \left(0 , \infty\right)$.

Note: This can be more concisely written as $D = \left\{\left(x , y\right) \in \boldsymbol{Z} \times \boldsymbol{Z} : y \ne 0\right\}$.