To determine the domain of the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], we need to understand the behavior of the cube root function.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers. This means that there are no restrictions on the input [tex]\( x \)[/tex] for the cube root function. In other words, [tex]\( \sqrt[3]{x} \)[/tex] can take any real number as an input and produce a real number as output.
Now, when considering the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], we are looking at a shift of the cube root function by 1 unit to the right. This shift does not affect the domain of the function in terms of the possible input values for [tex]\( x \)[/tex].
Since the cube root function is defined for all real numbers, and the shift [tex]\( x - 1 \)[/tex] is simply a translation of the input along the x-axis, the domain remains all real numbers.
Thus, the domain of [tex]\( y = \sqrt[3]{x - 1} \)[/tex] is all real numbers:
[tex]\[
-\infty < x < \infty
\][/tex]
Therefore, the correct option corresponding to this domain is:
[tex]\[
-\infty < x < \infty
\][/tex]
