To determine the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to identify all the possible values of [tex]\( x \)[/tex] for which the function is defined. The square root function, [tex]\( \sqrt{x} \)[/tex], has specific properties that restrict its domain.
1. Understanding the Square Root Function:
The square root function, [tex]\( \sqrt{x} \)[/tex], is only defined for non-negative numbers. This means that [tex]\( x \)[/tex] must be greater than or equal to zero. If [tex]\( x \)[/tex] is negative, [tex]\( \sqrt{x} \)[/tex] does not yield a real number.
2. Domain of [tex]\( y = \sqrt{x} \)[/tex]:
- The smallest value [tex]\( x \)[/tex] can take is 0. This is because [tex]\( \sqrt{0} = 0 \)[/tex] is defined.
- [tex]\( x \)[/tex] can take any value greater than 0, extending towards positive infinity ([tex]\( \infty \)[/tex]) since the square root of any non-negative number is defined in the real numbers.
Therefore, combining these conditions, the domain of [tex]\( y = \sqrt{x} \)[/tex] includes all non-negative real numbers, which can be written as:
[tex]\[ 0 \leq x < \infty \][/tex]
3. Comparing with Given Options:
- [tex]\( -\infty < x < \infty \)[/tex]: This includes negative values of [tex]\( x \)[/tex], which are not allowed for [tex]\( \sqrt{x} \)[/tex].
- [tex]\( 0 < x < \infty \)[/tex]: This excludes 0, but [tex]\( \sqrt{0} \)[/tex] is defined.
- [tex]\( 0 \leq x < \infty \)[/tex]: This includes 0 and all positive values of [tex]\( x \)[/tex], which matches our determined domain.
- [tex]\( 1 \leq x < \infty \)[/tex]: This excludes values between 0 and 1, including 0, which are part of the domain of [tex]\( \sqrt{x} \)[/tex].
Thus, the correct choice is:
[tex]\[ 0 \leq x < \infty \][/tex]
