To determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to identify all the possible values of [tex]\( x \)[/tex] for which the function is defined.
The function [tex]\( y = \sqrt[3]{x} \)[/tex] represents the cube root of [tex]\( x \)[/tex]. A cube root function has some important properties:
1. The cube root of any real number is always defined.
2. This means that you can take the cube root of a positive number, a negative number, or zero, and the result will still be a real number.
In other words:
- If [tex]\( x \)[/tex] is positive, [tex]\( \sqrt[3]{x} \)[/tex] will be a positive number.
- If [tex]\( x \)[/tex] is negative, [tex]\( \sqrt[3]{x} \)[/tex] will be a negative number.
- If [tex]\( x \)[/tex] is zero, [tex]\( \sqrt[3]{x} \)[/tex] will be zero.
Since there are no restrictions on [tex]\( x \)[/tex] for the function [tex]\( y = \sqrt[3]{x} \)[/tex], the set of all possible values of [tex]\( x \)[/tex] is all real numbers.
Therefore, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is [tex]\( -\infty < x < \infty \)[/tex].
The correct choice among the given options is:
[tex]\[ -\infty < x < \infty \][/tex]
