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What is the domain of the function [tex]y = \sqrt[3]{x-1}[/tex]?

A. [tex]-\infty \ \textless \ x \ \textless \ \infty[/tex]
B. [tex]-1 \ \textless \ x \ \textless \ \infty[/tex]
C. [tex]0 \leq x \ \textless \ \infty[/tex]
D. [tex]1 \leq x \ \textless \ \infty[/tex]

Sagot :

GinnyAnsweridn
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.
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