To determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to understand which values of [tex]\( x \)[/tex] can be input into the function to produce real number outputs.
1. Understanding Cube Roots:
- The cube root function, denoted as [tex]\(\sqrt[3]{x}\)[/tex] or [tex]\(x^{1/3}\)[/tex], is defined as the number that, when cubed, gives [tex]\( x \)[/tex].
- Unlike the square root function, the cube root function is defined for all real numbers, including negative numbers, zero, and positive numbers.
2. Analyzing Possible Values of [tex]\( x \)[/tex]:
- When [tex]\( x \)[/tex] is positive, [tex]\(\sqrt[3]{x}\)[/tex] is a positive real number. For example, [tex]\(\sqrt[3]{8} = 2\)[/tex].
- When [tex]\( x \)[/tex] is zero, [tex]\(\sqrt[3]{0}\)[/tex] is zero. For example, [tex]\(\sqrt[3]{0} = 0\)[/tex].
- When [tex]\( x \)[/tex] is negative, [tex]\(\sqrt[3]{x}\)[/tex] is a negative real number. For example, [tex]\(\sqrt[3]{-8} = -2\)[/tex].
3. Domain Conclusion:
- Since the cube root function can take any real number as input and produce a real number as output, there are no restrictions on the domain of [tex]\( x \)[/tex].
- Therefore, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is all real numbers.
Based on this analysis, the correct domain for the function [tex]\( y = \sqrt[3]{x} \)[/tex] is:
[tex]\[ -\infty < x < \infty \][/tex]
So the answer is:
[tex]\[ -\infty < x < \infty \][/tex]
