Question

what is the domain of $f(x) = \sqrt3{x}$?

all real numbers

positive numbers and zero

all integers

whole numbers

Explanation:

Step1: Recall the domain of cube root function

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 its cube root. For example, if \( x = -8 \), \( \sqrt[3]{-8}=-2 \); if \( x = 8 \), \( \sqrt[3]{8} = 2 \); if \( x = 0 \), \( \sqrt[3]{0}=0 \).

Step2: Analyze the options

• Option "positive numbers and zero" is incorrect because cube root is defined for negative numbers too.
• Option "all integers" is incorrect because the domain is not restricted to integers, it includes all real numbers (fractions, irrational numbers etc.).
• Option "whole numbers" is incorrect as whole numbers are non - negative integers, and the cube root function is defined for negative numbers and non - integer real numbers.
• Option "all real numbers" is correct as explained in step 1.

Answer:

A. all real numbers