To determine the domain of the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], we need to consider the values of [tex]\( x \)[/tex] for which the function is defined and yields real numbers.
The cube root function, [tex]\( \sqrt[3]{u} \)[/tex], is defined for all real numbers [tex]\( u \)[/tex]. This is because the cube root of any real number is also a real number. There are no restrictions or limitations on [tex]\( u \)[/tex] since the cube root function can handle negative numbers, zero, and positive numbers without any issues.
In our function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], the expression inside the cube root is [tex]\( x - 1 \)[/tex]. Since the cube root function can accept any real number as input, [tex]\( x - 1 \)[/tex] can also be any real number.
To put it succinctly:
- [tex]\( x - 1 \)[/tex] can be any real number.
- Therefore, [tex]\( x \)[/tex] can be any real number.
Thus, the domain of the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex] is all real numbers, represented as:
[tex]\[
-\infty < x < \infty
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{-\infty < x < \infty} \][/tex]
