To determine the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to identify the set of all possible values of [tex]\( x \)[/tex] for which the function is defined.
The square root function [tex]\( \sqrt{x} \)[/tex] requires the expression under the square root, [tex]\( x \)[/tex], to be non-negative because the principal square root of a negative number is not defined within the real numbers. Therefore, [tex]\( x \)[/tex] must be greater than or equal to zero.
Given this requirement, let's analyze the provided options:
1. [tex]\( -\infty < x < \infty \)[/tex]
This option suggests [tex]\( x \)[/tex] can be any real number, including negative numbers, which is incorrect because the square root function is not defined for negative values in the real number system.
2. [tex]\( 0 < x < \infty \)[/tex]
This option excludes zero but includes all positive real numbers. While the function is indeed defined for all positive [tex]\( x \)[/tex], it is also defined at [tex]\( x = 0 \)[/tex].
3. [tex]\( 0 \leq x < \infty \)[/tex]
This option includes zero and all positive real numbers. Since the function [tex]\( y = \sqrt{x} \)[/tex] is defined for [tex]\( x = 0 \)[/tex] (where [tex]\( \sqrt{0} = 0 \)[/tex]) and all positive [tex]\( x \)[/tex], this is the correct description of the domain.
4. [tex]\( 1 \leq x < \infty \)[/tex]
This option implies that [tex]\( x \)[/tex] must be at least 1, excluding zero and any values between 0 and 1. This is more restrictive than necessary and does not describe the full domain of the function.
Thus, the correct domain for the function [tex]\( y = \sqrt{x} \)[/tex] is [tex]\( 0 \leq x < \infty \)[/tex].
So, the correct answer is:
[tex]$ \boxed{0 \leq x < \infty} $[/tex]
The corresponding choice number to this domain specification is:
[tex]$ \boxed{3} $[/tex]
