Question

what is the domain of $f(x) = \sqrt3{x}$?
\bigcirc all real numbers
\bigcirc positive numbers and zero
\bigcirc all integers
\bigcirc whole numbers

Explanation:

Step1: Recall Domain of Cube Root

The cube root function \( f(x) = \sqrt[3]{x} \) is defined for all real numbers. Because for any real number \( x \) (positive, negative, or zero), we can find a real cube root. For example, \( \sqrt[3]{8} = 2 \), \( \sqrt[3]{-8} = -2 \), and \( \sqrt[3]{0} = 0 \).

Step2: Evaluate Options

• "positive numbers and zero": Excludes negative numbers, but cube root of negative numbers is defined (e.g., \( \sqrt[3]{-1} = -1 \)), so this is wrong.
• "all integers": The domain isn't restricted to integers; \( x \) can be any real number (e.g., \( \sqrt[3]{2} \) is valid), so this is wrong.
• "whole numbers": Whole numbers are non - negative integers, which is a subset of real numbers and too restrictive, so this is wrong.
• "all real numbers": Matches the domain of the cube root function.

Answer:

A. all real numbers