To determine the domain of the function \( y = \sqrt{x} + 4 \), we need to establish the set of all possible values for \( x \) that will make the function well-defined.
1. The function \( y \) consists of the square root of \( x \) plus 4.
2. To ensure that the square root is defined, the expression inside the square root, \( x \), must be non-negative. In other words, we require:
[tex]\[ x \geq 0 \][/tex]
By analyzing this condition, we can conclude that \( x \) needs to be greater than or equal to 0. This means that the variable \( x \) can take any value starting from 0 and extending to positive infinity.
Therefore, the domain of the function \( y = \sqrt{x} + 4 \) is:
[tex]\[ 0 \leq x < \infty \][/tex]
Among the given choices:
1. \(-\infty < x < \infty\) — Incorrect, as \( x \) cannot take negative values due to the square root.
2. \(-4 \leq x < \infty\) — Incorrect, as \( x \) starting from -4 does not make sense for a square root function.
3. \( 0 \leq x < \infty \) — Correct, as it matches the requirement for the square root function.
4. \( 4 \leq x < \infty \) — Incorrect, as it unnecessarily restricts \( x \) to start from 4.
Thus, the correct domain is \( 0 \leq x < \infty \), which corresponds to the third given option.
So, the answer is:
[tex]\[ \boxed{3} \][/tex]
