Answer :
To determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to understand the conditions under which the function is defined. The cube root function is defined for all real numbers. This is because the cube root of any real number [tex]\( x \)[/tex] is also a real number.
Let's go through the option choices one by one:
1. [tex]\(-\infty < x < \infty\)[/tex]:
- This means that [tex]\( x \)[/tex] can be any real number, which is a true statement for the cube root function.
2. [tex]\( 0 < x < \infty \)[/tex]:
- This means that [tex]\( x \)[/tex] can only be positive numbers, which would exclude zero and negative numbers. The cube root of zero and negative numbers are still defined.
3. [tex]\(0 \leq x < \infty\)[/tex]:
- This means [tex]\( x \)[/tex] can be zero or any positive number. It excludes negative numbers but the cube root of negative numbers is defined.
4. [tex]\( 1 \leq x < \infty \)[/tex]:
- This means [tex]\( x \)[/tex] must be at least 1 and includes positive numbers greater than or equal to 1, but excludes zero and negative numbers.
The correct domain for the function [tex]\( y = \sqrt[3]{x} \)[/tex] is all real numbers, because the cube root is defined for every real number. Therefore, the domain is [tex]\(-\infty < x < \infty\)[/tex].
Thus, the correct option is:
[tex]\[ -\infty < x < \infty \][/tex]
Let's go through the option choices one by one:
1. [tex]\(-\infty < x < \infty\)[/tex]:
- This means that [tex]\( x \)[/tex] can be any real number, which is a true statement for the cube root function.
2. [tex]\( 0 < x < \infty \)[/tex]:
- This means that [tex]\( x \)[/tex] can only be positive numbers, which would exclude zero and negative numbers. The cube root of zero and negative numbers are still defined.
3. [tex]\(0 \leq x < \infty\)[/tex]:
- This means [tex]\( x \)[/tex] can be zero or any positive number. It excludes negative numbers but the cube root of negative numbers is defined.
4. [tex]\( 1 \leq x < \infty \)[/tex]:
- This means [tex]\( x \)[/tex] must be at least 1 and includes positive numbers greater than or equal to 1, but excludes zero and negative numbers.
The correct domain for the function [tex]\( y = \sqrt[3]{x} \)[/tex] is all real numbers, because the cube root is defined for every real number. Therefore, the domain is [tex]\(-\infty < x < \infty\)[/tex].
Thus, the correct option is:
[tex]\[ -\infty < x < \infty \][/tex]
