To determine the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to identify the set of all possible values for [tex]\( x \)[/tex] that make the expression under the square root valid. The square root function is defined only for non-negative numbers because you cannot take the square root of a negative number in the real number system.
Here is the detailed step-by-step solution:
1. Understand the function [tex]\( y = \sqrt{x} \)[/tex]:
- The expression inside the square root, [tex]\( x \)[/tex], must be greater than or equal to zero (non-negative).
2. Set the constraint for [tex]\( x \)[/tex]:
- To keep [tex]\( \sqrt{x} \)[/tex] defined, [tex]\( x \)[/tex] must satisfy:
[tex]\[
x \geq 0
\][/tex]
3. Identify the interval for [tex]\( x \)[/tex]:
- The smallest value [tex]\( x \)[/tex] can take is 0.
- [tex]\( x \)[/tex] can be any value starting from [tex]\( 0 \)[/tex] and extending to positive infinity ([tex]\( \infty \)[/tex]).
4. Write the domain in interval notation:
- The domain of the function [tex]\( y = \sqrt{x} \)[/tex] is all values of [tex]\( x \)[/tex] from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], including 0.
- This can be written as:
[tex]\[
0 \leq x < \infty
\][/tex]
With this detailed reasoning, we can conclude that the domain of the function [tex]\( y = \sqrt{x} \)[/tex] is:
[tex]\[
0 \leq x < \infty
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{0 \leq x < \infty}
\][/tex]
