Question

what is the domain of the function $y = \sqrt3{x}$?
options:
$-\infty < x < \infty$
$0 \leq x < \infty$
$0 < x < \infty$
$1 \leq x < \infty$

Explanation:

Step1: Recall cube root domain

The cube root function \( y = \sqrt[3]{x} \) is defined for all real numbers. Because for any real number \( x \), the cube root \( \sqrt[3]{x} \) exists (e.g., \( \sqrt[3]{-8} = -2 \), \( \sqrt[3]{0} = 0 \), \( \sqrt[3]{8} = 2 \)). So the domain is all real numbers, which in interval notation is \( -\infty < x < \infty \).

Step2: Match with options

Among the given options, the one that represents all real numbers (from negative infinity to positive infinity) is \( -\infty < x < \infty \).

Answer:

\( -\infty < x < \infty \) (the option with \( -\infty < x < \infty \))