To determine the domain of the function \( f(x) = \sqrt{x + 3} + 4 \), we need to consider the conditions under which the expression under the square root is defined.
The square root function is defined for non-negative values. Therefore, we need:
\[ x + 3 \geq 0 \]
Solving this inequality:
\[ x \geq -3 \]
This means that the function \( f(x) \) is defined for all \( x \) greater than or equal to \(-3\).
In interval notation, the domain of the function \( f(x) \) is:
\[ [-3, \infty) \]
If you use graphing technology, you will observe that the graph of the function starts at the point \((-3, 4)\) and continues to increase as \( x \) increases, confirming that the function is defined for all \( x \) in the interval \([-3, \infty)\).