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 is because unlike the square root function, the cube root function can handle both positive and negative values. For instance:
- [tex]\( \sqrt[3]{8} = 2 \)[/tex]
- [tex]\( \sqrt[3]{-8} = -2 \)[/tex]
- [tex]\( \sqrt[3]{0} = 0 \)[/tex]
2. There are no restrictions or values of [tex]\( x \)[/tex] that would make the cube root function undefined. Whether [tex]\( x \)[/tex] is positive, negative, or zero, [tex]\( \sqrt[3]{x} \)[/tex] will always produce a real number.
Therefore, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] includes all real numbers from negative infinity to positive infinity.
Thus, the correct answer is:
[tex]\[ -\infty < x < \infty \][/tex]
