Connect with knowledgeable individuals and get your questions answered on IDNLearn.com. Our platform is designed to provide reliable and thorough answers to all your questions, no matter the topic.

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]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.