To determine the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex], we need to consider the nature of the cube root function.
The cube root function, denoted as [tex]\( \sqrt[3]{x} \)[/tex] or [tex]\( x^{1/3} \)[/tex], is defined for all real numbers. This means that you can take the cube root of any real number, positive or negative, and still obtain a real number as a result.
Given the function [tex]\( y = \sqrt[3]{x-1} \)[/tex]:
1. The expression inside the cube root is [tex]\( x - 1 \)[/tex].
2. Since the cube root function can accept any real number, [tex]\( x - 1 \)[/tex] can be any real number.
3. As a result, we do not need to place any specific restrictions on [tex]\( x \)[/tex]. We simply need [tex]\( x - 1 \)[/tex] to be a real number, which [tex]\( x \)[/tex] can always satisfy.
Thus, the domain of [tex]\( y = \sqrt[3]{x-1} \)[/tex] includes all real numbers [tex]\( x \)[/tex]. Therefore, the domain can be expressed as:
[tex]\[ -\infty < x < \infty \][/tex]
Thus, the correct option is:
[tex]\[ -\infty < x < \infty \][/tex]
