To determine the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex], let's analyze the properties of the cubic root function.
### Step-by-Step Analysis:
1. Understanding the Cubic Root Function:
- The function [tex]\( y = \sqrt[3]{x} \)[/tex] is the cubic root of [tex]\( x \)[/tex]. This means for any value of [tex]\( x \)[/tex], [tex]\( y \)[/tex] is the number which, when cubed, gives [tex]\( x \)[/tex].
- Mathematically, this can be expressed as [tex]\( y = x^{1/3} \)[/tex].
2. Consideration of Real Numbers:
- The cubic root function is defined for all real numbers. That means you can take the cubic root of any real number, whether it is positive, negative, or zero.
- For example:
- [tex]\( \sqrt[3]{8} = 2 \)[/tex] because [tex]\( 2^3 = 8 \)[/tex].
- [tex]\( \sqrt[3]{-27} = -3 \)[/tex] because [tex]\( (-3)^3 = -27 \)[/tex].
- [tex]\( \sqrt[3]{0} = 0 \)[/tex] because [tex]\( 0^3 = 0 \)[/tex].
3. Checking the Choices:
- Choice 1: [tex]\( -\infty < x < \infty \)[/tex]
- This represents all real numbers, which is consistent with the domain of the cubic root function.
- Choice 2: [tex]\( 0 < x < \infty \)[/tex]
- This represents only positive real numbers, which excludes zero and negative numbers.
- Choice 3: [tex]\( 0 \leq x < \infty \)[/tex]
- This includes zero and all positive numbers, but still excludes negative numbers.
- Choice 4: [tex]\( 1 \leq x < \infty \)[/tex]
- This includes only numbers greater than or equal to one, excluding zero and negative numbers.
4. Conclusion:
- The only choice that includes all real numbers, which is the correct domain for the cubic root function, is:
[tex]\[
-\infty < x < \infty
\][/tex]
Therefore, the domain of the function [tex]\( y = \sqrt[3]{x} \)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
