Question

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

Explanation:

Step1: Recall cube root domain rule

The cube root function \( \sqrt[3]{u} \) is defined for all real numbers \( u \), because we can take the cube root of any real number (positive, negative, or zero).

Step2: Analyze the function's inside

In the function \( y = \sqrt[3]{x - 1} \), the expression inside the cube root is \( x - 1 \). Since \( x - 1 \) is a real number for any real number \( x \) (because subtraction of real numbers is defined for all real numbers), there are no restrictions on \( x \) from the cube root operation.

Step3: Determine the domain

So the domain of \( y=\sqrt[3]{x - 1} \) is all real numbers, which is written as \( -\infty < x < \infty \).

Answer:

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