To determine the domain of the function [tex]\( y = \sqrt{x} + 4 \)[/tex], we need to analyze where the function is defined and produces real numbers.
1. The function [tex]\( y = \sqrt{x} + 4 \)[/tex] involves a square root operation. Recall that the square root function [tex]\(\sqrt{x}\)[/tex] is only defined for non-negative values of [tex]\( x \)[/tex], i.e., [tex]\( x \geq 0 \)[/tex].
2. Given this, we need to ensure that the expression inside the square root is always non-negative. For [tex]\( \sqrt{x} \)[/tex], this means [tex]\( x \)[/tex] must satisfy:
[tex]\[
x \geq 0
\][/tex]
3. Since [tex]\( \sqrt{x} \)[/tex] is non-negative for all [tex]\( x \geq 0 \)[/tex] and will produce real numbers, and adding 4 to [tex]\(\sqrt{x}\)[/tex] does not affect the domain restrictions further, we can conclude that the function [tex]\( y = \sqrt{x} + 4 \)[/tex] is defined for all [tex]\( x \)[/tex] in the interval:
[tex]\[
0 \leq x < \infty
\][/tex]
4. Thus, the domain of the function [tex]\( y = \sqrt{x} + 4 \)[/tex] is from 0 to infinity.
Out of the given options, the correct choice that represents this domain is:
[tex]\[
0 \leq x < \infty
\][/tex]
Therefore, the domain of the function [tex]\( y = \sqrt{x} + 4 \)[/tex] is [tex]\( 0 \leq x < \infty \)[/tex] and it corresponds to the third choice in the given options.
