To determine the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex], let's analyze it step by step.
1. Understanding the Function:
The function is a cube root, [tex]\( \sqrt[3]{x-1} \)[/tex]. A cube root function, unlike a square root function, is defined for all real numbers. This means you can take the cube root of any real number, including negative numbers.
2. Determining the Argument of the Function:
The expression inside the cube root is [tex]\( x - 1 \)[/tex]. For the cube root to be defined, [tex]\( x - 1 \)[/tex] must be any real number. There are no restrictions on [tex]\( x - 1 \)[/tex] because the cube root function can handle any real number input.
3. Translating to the Original Variable:
Since [tex]\( x - 1 \)[/tex] can indeed be any real number, we can conclude that [tex]\( x \)[/tex] itself can also be any real number. There is no value of [tex]\( x \)[/tex] that makes [tex]\( y \)[/tex] undefined.
4. Conclusion:
Therefore, the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex] is all real numbers, which is expressed as:
[tex]\[
-\infty < x < \infty
\][/tex]
So, the correct choice is:
[tex]\[
-\infty < x < \infty
\][/tex]
