To determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to identify all possible values of [tex]\( x \)[/tex] for which the function is defined.
1. The cube root function, [tex]\( \sqrt[3]{x} \)[/tex], is defined for all real numbers. This means:
- You can take the cube root of any positive number.
- You can take the cube root of zero.
- You can take the cube root of any negative number.
2. Since the cube root function does not have any restrictions on [tex]\( x \)[/tex] — it is defined for every real number — the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] includes all real numbers.
Given the options:
- [tex]\(-\infty < x < \infty\)[/tex]
- [tex]\(0 < x < \infty\)[/tex]
- [tex]\(0 \leq x < \infty\)[/tex]
- [tex]\(1 \leq x < \infty\)[/tex]
The correct option that represents the domain of [tex]\( y = \sqrt[3]{x} \)[/tex] is:
[tex]\[ -\infty < x < \infty \][/tex]
Thus, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is all real numbers.
