To determine the domain of the function [tex]\(f(x) = \sqrt[3]{x}\)[/tex], we need to identify all the possible values of [tex]\(x\)[/tex] for which the function is defined.
The cube root function, [tex]\( \sqrt[3]{x} \)[/tex] (or equivalently [tex]\( x^{\frac{1}{3}} \)[/tex]), is defined as the value that, when cubed, gives [tex]\(x\)[/tex]. In mathematical terms, for any real number [tex]\(a\)[/tex]:
[tex]\[ \sqrt[3]{a} = b \text{ such that } b^3 = a \][/tex]
### Characteristics of the Cube Root Function
1. Positive Inputs:
- For any positive real number [tex]\(x\)[/tex], the cube root is defined and positive. For example, [tex]\(\sqrt[3]{8} = 2\)[/tex], because [tex]\(2^3 = 8\)[/tex].
2. Zero:
- The cube root of zero is zero. For example, [tex]\(\sqrt[3]{0} = 0\)[/tex], because [tex]\(0^3 = 0\)[/tex].
3. Negative Inputs:
- For any negative real number [tex]\(x\)[/tex], the cube root is defined and negative. For example, [tex]\(\sqrt[3]{-8} = -2\)[/tex], because [tex]\((-2)^3 = -8\)[/tex].
### Conclusion
The cube root function is defined for all real numbers. There are no restrictions on the values [tex]\(x\)[/tex] can take for [tex]\( f(x) = \sqrt[3]{x} \)[/tex] to be defined.
Thus, the domain of [tex]\(f(x) = \sqrt[3]{x}\)[/tex] is all real numbers.
Among the given options, the correct answer is:
- all real numbers
The other options (positive numbers and zero, all integers, whole numbers) are incorrect since they do not include all possible real numbers which the cube root function can take.
