To determine the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to consider the conditions under which the expression inside the square root is defined.
1. Understanding the Square Root Function:
- The square root function is defined for any non-negative number.
- That means [tex]\( \sqrt{x} \)[/tex] is only defined when [tex]\( x \geq 0 \)[/tex].
2. Setting Up Inequalities:
- Since [tex]\( x \)[/tex] must be greater than or equal to 0 to avoid taking the square root of a negative number, we can set up the inequality:
[tex]\[ x \geq 0. \][/tex]
3. Expressing the Domain:
- The domain of a function is the set of all possible input values (in this case, [tex]\( x \)[/tex]) for which the function is defined.
- From the inequality, we can see that [tex]\( x \)[/tex] can take any value from 0 to positive infinity, inclusive.
- This can be written in interval notation as [tex]\( [0, \infty) \)[/tex].
4. Matching with Given Options:
- Let’s compare this interval with the given choices:
1. [tex]\(-\infty < x < \infty\)[/tex]: This includes negative values, which are not allowed for the square root function.
2. [tex]\(0 < x < \infty\)[/tex]: This does not include [tex]\( x = 0 \)[/tex], which is allowed.
3. [tex]\( 0 \leq x < \infty \)[/tex]: This includes all [tex]\( x \)[/tex] values from 0 to positive infinity, inclusive, which is correct.
4. [tex]\( 1 \leq x < \infty \)[/tex]: This does not include values between 0 and 1, which are allowed for the square root function.
Based on this analysis, the correct domain for the function [tex]\( y = \sqrt{x} \)[/tex] is [tex]\( 0 \leq x < \infty \)[/tex], and the corresponding option is:
[tex]\[ \boxed{3} \][/tex]
