To determine the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to consider the properties of the square root function.
The square root function is defined for all non-negative real numbers. This means that the value inside the square root (i.e., [tex]\( x \)[/tex]) must be greater than or equal to zero. In other words, [tex]\( \sqrt{x} \)[/tex] is defined if and only if [tex]\( x \geq 0 \)[/tex].
Let’s now analyze the four options given to see which one correctly describes this condition:
1. [tex]\( -\infty < x < \infty \)[/tex]: This option suggests that [tex]\( x \)[/tex] can be any real number, including negative values. However, [tex]\( \sqrt{x} \)[/tex] is not defined for negative values of [tex]\( x \)[/tex]. Therefore, this option is incorrect.
2. [tex]\( 0 < x < \infty \)[/tex]: This option suggests that [tex]\( x \)[/tex] must be positive but not zero. However, [tex]\( \sqrt{x} \)[/tex] is defined for [tex]\( x = 0 \)[/tex] because [tex]\( \sqrt{0} = 0 \)[/tex]. Therefore, this option is not entirely correct because it excludes zero.
3. [tex]\( 0 \leq x < \infty \)[/tex]: This option includes all non-negative real numbers, starting from zero and extending to infinity. Since [tex]\( \sqrt{x} \)[/tex] is defined for all [tex]\( x \)[/tex] in this range, this option correctly represents the domain of the function.
4. [tex]\( 1 \leq x < \infty \)[/tex]: This option suggests that [tex]\( x \)[/tex] must be greater than or equal to 1. This excludes all values from 0 to 1, which are part of the domain of [tex]\( \sqrt{x} \)[/tex]. Therefore, this option is incorrect.
Thus, the correct domain of the function [tex]\( y = \sqrt{x} \)[/tex] is [tex]\( 0 \leq x < \infty \)[/tex].
The correct answer is:
[tex]\[ \boxed{0 \leq x < \infty} \][/tex]
