Let's analyze the function [tex]\( y = \sqrt[3]{x} \)[/tex] in order to determine its domain.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is the inverse of the cube function [tex]\( x^3 \)[/tex]. An important property of the cube root function is that it is defined for all real numbers. This means that you can take the cube root of any real number, whether it is positive, negative, or zero.
For example:
- The cube root of a positive number [tex]\( \sqrt[3]{8} = 2 \)[/tex].
- The cube root of zero [tex]\( \sqrt[3]{0} = 0 \)[/tex].
- The cube root of a negative number [tex]\( \sqrt[3]{-8} = -2 \)[/tex].
Since there are no restrictions on the values that [tex]\( x \)[/tex] can take for the function [tex]\( y = \sqrt[3]{x} \)[/tex], the domain includes all real numbers.
Thus, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is:
[tex]\[ -\infty < x < \infty \][/tex]
Therefore, the correct answer is:
[tex]\[ -\infty < x < \infty \][/tex]
