To determine the domain of the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], we need to understand the nature of the cube root function.
The cube root function [tex]\( y = \sqrt[3]{z} \)[/tex] is defined for all real numbers [tex]\( z \)[/tex]. This is because the cube root of any real number is also a real number. There are no restrictions on the input [tex]\( z \)[/tex] when taking its cube root.
Given the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], we can rewrite it in the form [tex]\( y = \sqrt[3]{z} \)[/tex] by letting [tex]\( z = x - 1 \)[/tex]. Since [tex]\( z \)[/tex] can be any real number, [tex]\( x - 1 \)[/tex] can also be any real number. Therefore, [tex]\( x \)[/tex] itself can be any real number.
In other words, there are no restrictions on [tex]\( x \)[/tex] for the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex]. The function is defined for all real values of [tex]\( x \)[/tex].
Thus, the domain of the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex] is:
[tex]\[
-\infty < x < \infty
\][/tex]
So, the correct answer is:
[tex]\[
-\infty < x < \infty
\][/tex]
