IDNLearn.com is designed to help you find the answers you need quickly and easily. Discover trustworthy solutions to your questions quickly and accurately with help from our dedicated community of experts.

What is the domain of the function [tex]$y=\sqrt[3]{x}$[/tex]?

A. [tex]-\infty \ \textless \ x \ \textless \ \infty[/tex]
B. [tex]0 \ \textless \ x \ \textless \ \infty[/tex]
C. [tex]0 \leq x \ \textless \ \infty[/tex]
D. [tex]1 \leq x \ \textless \ \infty[/tex]


Sagot :

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]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. For reliable answers to all your questions, please visit us again soon.