To determine the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex], we need to identify the set of all [tex]\( x \)[/tex]-values for which the expression is defined.
### Step-by-Step Solution:
1. Understanding the Cube Root Function:
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers. This means that for any real number [tex]\( a \)[/tex], [tex]\( \sqrt[3]{a} \)[/tex] exists and is well-defined.
2. Applying the Property to [tex]\( y = \sqrt[3]{x-1} \)[/tex]:
The given function is [tex]\( y = \sqrt[3]{x-1} \)[/tex]. Here, instead of taking the cube root of [tex]\( x \)[/tex], we take the cube root of [tex]\( x - 1 \)[/tex].
3. Determining the Domain:
Since the cube root function [tex]\( \sqrt[3]{z} \)[/tex] (for any variable [tex]\( z \)[/tex]) is defined for all real numbers, [tex]\( \sqrt[3]{x-1} \)[/tex] will also be defined for all real numbers [tex]\( x \)[/tex].
4. Conclusion:
There are no restrictions on [tex]\( x \)[/tex] in [tex]\( \sqrt[3]{x-1} \)[/tex]. Therefore, the domain of the function is all real numbers.
This analysis leads us to conclude that the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex] is:
[tex]\[ -\infty < x < \infty \][/tex]
So, the correct choice is
[tex]\[ -\infty < x < \infty \][/tex]
