To find the domain of the function \( f(x) = \sqrt{x + 2} + 2 \), we need to determine the values of \( x \) for which the function is defined.
The square root function \( \sqrt{x + 2} \) is defined only when the expression inside the square root is non-negative. Therefore, we need:
\[ x + 2 \geq 0 \]
Solving this inequality:
\[ x \geq -2 \]
Thus, the domain of the function \( f(x) = \sqrt{x + 2} + 2 \) is all values of \( x \) greater than or equal to \(-2\). In interval notation, the domain is:
\[ [-2, \infty) \]
Summary
The domain of the function \( f(x) = \sqrt{x + 2} + 2 \) is \([-2, \infty)\).