To determine the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex], we need to identify all the possible values of [tex]\( x \)[/tex] for which the function is defined.
### Step-by-step Analysis:
1. Understand the Function:
- The given function is [tex]\( y = \sqrt[3]{x-1} \)[/tex], which represents the cube root of [tex]\( x-1 \)[/tex].
2. Properties of the Cube Root Function:
- The cube root function, [tex]\( \sqrt[3]{u} \)[/tex], is defined for all real numbers [tex]\( u \)[/tex].
- In other words, you can take the cube root of any real number, including negative, positive, and zero values without any restrictions.
3. Determine the Argument [tex]\( x-1 \)[/tex]:
- Here, the argument inside the cube root is [tex]\( x-1 \)[/tex]. Since the cube root of any real number [tex]\( x-1 \)[/tex] is defined, [tex]\( x-1 \)[/tex] can be any real number.
4. No Restrictions:
- There are no restrictions on the values that [tex]\( x \)[/tex] can take because subtracting 1 from any real number still produces a real number, and the cube root of that result is also a real number.
### Conclusion:
- Based on these observations, [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, which can be written in interval notation as:
[tex]\[ (-\infty, \infty) \][/tex]
So, the correct answer is:
[tex]\[ -\infty < x < \infty \][/tex]
