To find the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to determine for which values of [tex]\( x \)[/tex] the expression under the square root is defined and produces a real number.
1. Square Root Function Property: The square root function [tex]\( \sqrt{x} \)[/tex] is defined for all non-negative values of [tex]\( x \)[/tex]. This is because the square root of a negative number is not a real number; it is an imaginary number.
2. Non-negative [tex]\( x \)[/tex]: Therefore, [tex]\( x \)[/tex] must be greater than or equal to 0 (i.e., [tex]\( x \geq 0 \)[/tex]). Any value of [tex]\( x \)[/tex] that is less than 0 would make the expression [tex]\( \sqrt{x} \)[/tex] undefined in the real number system.
3. Not Including Negative Values: Because of this property, negative values for [tex]\( x \)[/tex] like [tex]\( -1, -2, -3, \ldots \)[/tex] are excluded from the domain.
4. Including Zero: On the other hand, [tex]\( \sqrt{0} \)[/tex] is defined and equals 0, so [tex]\( x \)[/tex] can be equal to 0.
5. Unbounded Upper Limit: Since [tex]\( x \)[/tex] can take any non-negative value extending towards infinity, the upper limit is positive infinity.
Hence, combining these criteria, the domain of the function [tex]\( y = \sqrt{x} \)[/tex] is all [tex]\( x \)[/tex] that satisfy [tex]\( x \geq 0 \)[/tex]. In interval notation, this is represented as:
[tex]\[ [0, \infty) \][/tex]
Comparing this with the given options:
- [tex]\(-\infty < x < \infty\)[/tex] includes negative values, which is incorrect.
- [tex]\(0 < x < \infty\)[/tex] excludes [tex]\( x = 0 \)[/tex], which is incorrect.
- [tex]\(0 \leq x < \infty\)[/tex] includes [tex]\( x = 0 \)[/tex] and all positive [tex]\( x \)[/tex], which is correct.
- [tex]\(1 \leq x < \infty\)[/tex] starts from [tex]\( x = 1 \)[/tex], which is incorrect.
Therefore, the correct answer is:
[tex]\[ 0 \leq x < \infty \][/tex]
