To determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to understand when the function [tex]\( \sqrt[3]{x} \)[/tex] is defined.
1. Understanding cube roots: The cube root function, written as [tex]\( \sqrt[3]{x} \)[/tex] or equivalently [tex]\( x^{1/3} \)[/tex], takes any real number [tex]\( x \)[/tex] and returns a real number [tex]\( y \)[/tex] such that [tex]\( y^3 = x \)[/tex].
2. Characteristics of cube roots: Unlike square roots which require non-negative inputs to produce real numbers, cube roots can accept both positive and negative inputs. For example:
- [tex]\( \sqrt[3]{8} = 2 \)[/tex] because [tex]\( 2^3 = 8 \)[/tex]
- [tex]\( \sqrt[3]{-8} = -2 \)[/tex] because [tex]\( (-2)^3 = -8 \)[/tex]
- [tex]\( \sqrt[3]{0} = 0 \)[/tex] because [tex]\( 0^3 = 0 \)[/tex]
3. Set of possible inputs: Because the cube root function can take both negative and positive numbers as well as zero, there is no restriction on [tex]\( x \)[/tex]. Hence, [tex]\( x \)[/tex] can be any real number.
4. Domain in interval notation: Considering that all real numbers are included within the domain, we can write the domain as the interval [tex]\( (-\infty, \infty) \)[/tex].
Among the given options, the option [tex]\( -\infty < x < \infty \)[/tex] is the correct one. Therefore, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is:
[tex]\[ -\infty < x < \infty \][/tex]
