To determine the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex], we need to analyze for which values of [tex]\( x \)[/tex] the expression inside the cube root is defined.
The function [tex]\( y = \sqrt[3]{x-1} \)[/tex] is a cube root function. Unlike square root functions, which are only defined for non-negative inputs, cube root functions are defined for all real numbers. This is because a cube root of any real number (whether positive, negative, or zero) is a real number.
Let's look at the function step-by-step:
1. The function is [tex]\( y = \sqrt[3]{x - 1} \)[/tex].
2. Inside the cube root, we have [tex]\( x - 1 \)[/tex].
For the cube root to be defined, [tex]\( x - 1 \)[/tex] can be any real number. There are no restrictions because:
- The cube root of a positive number is defined.
- The cube root of zero is defined.
- The cube root of a negative number is defined.
Therefore, there are no restrictions on [tex]\( x \)[/tex]. This means that [tex]\( x \)[/tex] can take any value from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
Thus, 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]
