To determine the domain of the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], we need to understand for which values of [tex]\( x \)[/tex] the function is defined.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers. This is because the cube root of a number, whether positive, negative, or zero, always exists.
Here's a more detailed explanation:
1. Positive Numbers: For any positive [tex]\( x \)[/tex], [tex]\( \sqrt[3]{x} \)[/tex] will yield a positive result. For example, [tex]\( \sqrt[3]{8} = 2 \)[/tex].
2. Negative Numbers: For any negative [tex]\( x \)[/tex], [tex]\( \sqrt[3]{x} \)[/tex] will yield a negative result. For example, [tex]\( \sqrt[3]{-8} = -2 \)[/tex].
3. Zero: The cube root of zero is zero, [tex]\( \sqrt[3]{0} = 0 \)[/tex].
Therefore, since [tex]\( \sqrt[3]{x} \)[/tex] has a well-defined output for every real number [tex]\( x \)[/tex], the domain of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is all real numbers.
Thus, the correct answer is:
- all real numbers
