To determine the domain of the function [tex]\( y = \sqrt{x} \)[/tex], we need to identify the set of all possible [tex]\( x \)[/tex]-values for which the expression under the square root is defined.
1. The square root function [tex]\( \sqrt{x} \)[/tex] is defined only for non-negative numbers, because you cannot take the square root of a negative number and get a real result.
2. Therefore, the expression under the square root, [tex]\( x \)[/tex], must be greater than or equal to 0. This gives us the inequality:
[tex]\[
x \geq 0
\][/tex]
3. In interval notation, this is written as:
[tex]\[
[0, \infty)
\][/tex]
Symbolically, this can also be expressed as:
[tex]\[
0 \leq x < \infty
\][/tex]
Let's evaluate each of the given options:
1. [tex]\( -\infty < x < \infty \)[/tex]:
- This option includes negative numbers, which are not in the domain of [tex]\( y = \sqrt{x} \)[/tex]. So, this is incorrect.
2. [tex]\( 0 < x < \infty \)[/tex]:
- This option excludes [tex]\( x = 0 \)[/tex]. However, because [tex]\( \sqrt{0} = 0 \)[/tex] is defined, we should include [tex]\( x = 0 \)[/tex] in the domain. So, this is incorrect.
3. [tex]\( 0 \leq x < \infty \)[/tex]:
- This option includes all non-negative numbers and matches our reasoning above. This is the correct domain.
4. [tex]\( 1 \leq x < \infty \)[/tex]:
- This option excludes numbers between 0 and 1, which are in the domain of [tex]\( y = \sqrt{x} \)[/tex]. So, this is incorrect.
Therefore, the correct domain of the function [tex]\( y = \sqrt{x} \)[/tex] is:
[tex]\[
0 \leq x < \infty
\][/tex]
The corresponding option is:
[tex]\[
3
\][/tex]
