To find the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we must determine the set of all possible values of [tex]\( x \)[/tex] for which the expression under the square root is defined and real.
1. Understanding the square root function:
- The square root function [tex]\( y = \sqrt{x} \)[/tex] returns real numbers only for non-negative values of [tex]\( x \)[/tex], as the square root of a negative number is not a real number.
2. Non-negative values of [tex]\( x \)[/tex]:
- Therefore, [tex]\( \sqrt{x} \)[/tex] is defined if and only if [tex]\( x \)[/tex] is greater than or equal to 0. Mathematically, this means [tex]\( x \geq 0 \)[/tex].
3. Expressing the domain:
- In interval notation, this can be written as [tex]\( 0 \leq x < \infty \)[/tex]. This interval includes all real numbers starting from 0 and extending towards positive infinity.
By analyzing the options provided:
1. [tex]\( -\infty < x < \infty \)[/tex]: This includes all real numbers, both negative and positive, which is incorrect since [tex]\( x \)[/tex] must be non-negative.
2. [tex]\( 0 < x < \infty \)[/tex]: This only includes positive real numbers but excludes 0, which is incorrect since 0 is part of the domain.
3. [tex]\( 0 \leq x < \infty \)[/tex]: This includes all non-negative real numbers starting from 0, which is correct.
4. [tex]\( 1 \leq x < \infty \)[/tex]: This only includes positive real numbers starting from 1, which is incorrect since both 0 and numbers between 0 and 1 are excluded.
Thus, the correct domain of the function [tex]\( y = \sqrt{x} \)[/tex] is:
[tex]\[ 0 \leq x < \infty \][/tex]
This corresponds to the third option.
Therefore, the answer is:
[tex]\[ \boxed{3} \][/tex]
