To determine the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex], we need to understand the properties of the cube root function. The cube root function, denoted as [tex]\( \sqrt[3]{x} \)[/tex], is defined for all real numbers. This means we can take the cube root of any real number without restriction.
Let's analyze this step-by-step:
1. The function [tex]\( y = \sqrt[3]{x-1} \)[/tex] involves shifting the cube root function horizontally. Specifically, it shifts the function to the right by 1 unit.
- For the cube root function [tex]\( y = \sqrt[3]{x} \)[/tex], the domain is all real numbers, [tex]\( (-\infty, \infty) \)[/tex], because you can cube root any real number and get a real result.
- When we replace [tex]\( x \)[/tex] with [tex]\( x-1 \)[/tex], we are translating the graph of the cube root function one unit to the right. This translation does not limit the values that [tex]\( x \)[/tex] can take. Hence, [tex]\( x-1 \)[/tex] can be any real number as well.
2. Consequently, since the cube root function is defined for all real numbers, and a horizontal shift does not restrict the input values, the domain of [tex]\( y = \sqrt[3]{x-1} \)[/tex] is the set of all real numbers.
Therefore, the domain of the function [tex]\( y = \sqrt[3]{x-1} \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
The most appropriate choice based on the given options is:
[tex]\[ -\infty < x < \infty \][/tex]
So, the correct answer is:
The domain is [tex]\( (-\infty, \infty) \)[/tex].
