To determine the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex], we need to understand the behavior of cube root functions in general.
### Understanding Cube Root Functions
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Unlike square root functions, which are only defined for non-negative values of [tex]\( x \)[/tex], cube root functions can take any real number input, including negative values. This is because the cube root of a negative number is also a real number.
### Analyzing [tex]\( y = \sqrt[3]{x-1} \)[/tex]
For the given function [tex]\( y = \sqrt[3]{x-1} \)[/tex]:
1. Expression Analysis: The expression inside the cube root is [tex]\( x - 1 \)[/tex].
2. Domain Determination: Since any real number [tex]\( x - 1 \)[/tex] can be substituted into a cube root function and yield a real result, there is no restriction on [tex]\( x \)[/tex].
### Conclusion
Thus, [tex]\( x \)[/tex] can be any real number. Therefore, the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex] is:
[tex]\[ -\infty < x < \infty \][/tex]
Hence, the correct answer is:
[tex]\[ -\infty < x < \infty \][/tex]
Which corresponds to the first option.
