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]



Answer :

To determine the domain of the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], we need to understand the behavior of the cube root function.

The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers. This means that there are no restrictions on the input [tex]\( x \)[/tex] for the cube root function. In other words, [tex]\( \sqrt[3]{x} \)[/tex] can take any real number as an input and produce a real number as output.

Now, when considering the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex], we are looking at a shift of the cube root function by 1 unit to the right. This shift does not affect the domain of the function in terms of the possible input values for [tex]\( x \)[/tex].

Since the cube root function is defined for all real numbers, and the shift [tex]\( x - 1 \)[/tex] is simply a translation of the input along the x-axis, the domain remains all real numbers.

Thus, the domain of [tex]\( y = \sqrt[3]{x - 1} \)[/tex] is all real numbers:

[tex]\[ -\infty < x < \infty \][/tex]

Therefore, the correct option corresponding to this domain is:

[tex]\[ -\infty < x < \infty \][/tex]