To determine the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to identify all the values of [tex]\( x \)[/tex] for which the function is defined.
The function [tex]\( y = \sqrt{x} \)[/tex] entails taking the square root of [tex]\( x \)[/tex]. There are important properties of square roots that we need to consider:
1. The square root of a non-negative number [tex]\( x \)[/tex] (where [tex]\( x \geq 0 \)[/tex]) is defined.
2. The square root of a negative number is not defined in the realm of real numbers (it would yield an imaginary number).
Given this information, we can conclude that:
- [tex]\( x \)[/tex] must be greater than or equal to 0.
- There is no upper limit for [tex]\( x \)[/tex], meaning [tex]\( x \)[/tex] can extend to infinity in the positive direction.
Combining these observations, the domain of the function [tex]\( y = \sqrt{x} \)[/tex] is [tex]\( 0 \leq x < \infty \)[/tex].
Among the given options:
- [tex]\( -\infty < x < \infty \)[/tex] includes negative numbers, which are not part of the domain.
- [tex]\( 0 < x < \infty \)[/tex] excludes 0, although 0 is a valid input for the function [tex]\( y = \sqrt{x} \)[/tex] since [tex]\( \sqrt{0} = 0 \)[/tex].
- [tex]\( 0 \leq x < \infty \)[/tex] correctly includes all non-negative numbers starting from 0 and extending to infinity.
- [tex]\( 1 \leq x < \infty \)[/tex] excludes 0 and other positive numbers less than 1, which are valid inputs for [tex]\( y = \sqrt{x} \)[/tex].
Thus, the domain of the function [tex]\( y = \sqrt{x} \)[/tex] is best described by the choice [tex]\( 0 \leq x < \infty \)[/tex].
Therefore, the correct answer is:
[tex]\[ 0 \leq x < \infty \][/tex]
