Let's determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex].
The domain of a function consists of all the possible input values (x-values) for which the function is defined and produces a valid output.
For the cube root function [tex]\( y = \sqrt[3]{x} \)[/tex], let's consider the properties:
1. The cube root of any real number is defined. In other words, you can take the cube root of both positive and negative numbers, as well as zero:
- [tex]\(\sqrt[3]{8} = 2\)[/tex]
- [tex]\(\sqrt[3]{-8} = -2\)[/tex]
- [tex]\(\sqrt[3]{0} = 0\)[/tex]
2. There are no restrictions on the values of [tex]\( x \)[/tex] that can be input into the cube root function. Unlike the square root function, the cube root function is defined for all real numbers, including negative numbers.
3. Because the cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] is defined for all real numbers, the input [tex]\( x \)[/tex] can take any value from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
Based on this reasoning, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is:
[tex]\[
-\infty < x < \infty
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{-\infty < x < \infty}
\][/tex]
