To determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to analyze the behavior of the cube root function and find the set of all possible input values [tex]\( x \)[/tex] for which the function is defined.
1. The cube root function, denoted as [tex]\( \sqrt[3]{x} \)[/tex] or [tex]\( x^{1/3} \)[/tex], is defined for all real numbers. This includes both positive and negative numbers as well as zero.
2. Unlike the square root function (which is only defined for non-negative numbers), the cube root function does not have any restrictions based on the sign of [tex]\( x \)[/tex]. In other words, [tex]\( \sqrt[3]{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex] as well as [tex]\( x < 0 \)[/tex].
This can be understood by considering a few examples:
- For [tex]\( x = 27 \)[/tex], [tex]\( \sqrt[3]{27} = 3 \)[/tex].
- For [tex]\( x = -8 \)[/tex], [tex]\( \sqrt[3]{-8} = -2 \)[/tex].
- For [tex]\( x = 0 \)[/tex], [tex]\( \sqrt[3]{0} = 0 \)[/tex].
These examples confirm that [tex]\( y = \sqrt[3]{x} \)[/tex] is defined for any real number [tex]\( x \)[/tex]. Thus, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is all real numbers.
Therefore, the domain can be represented as [tex]\( -\infty < x < \infty \)[/tex], which means the function is defined for all [tex]\( x \)[/tex] in the set of real numbers.
So the correct choice is:
[tex]\[ -\infty < x < \infty \][/tex]
