1. Problem: Find the domain and range of the function $f(x) = \sqrt{x^2 + 4}$. 2. Formula and rules: The domain of a function under a square root must satisfy the radicand $\geq 0$. 3. Domain: Since $x^2 + 4 \geq 4 > 0$ for all real $x$, the domain is all real numbers: $\mathbb{R}$. 4. Range: The smallest value inside the root is 4, so the smallest value of $f(x)$ is $\sqrt{4} = 2$. As $|x|$ grows large, $x^2$ dominates and $f(x) \to \infty$. Thus, range is $[2, \infty)$. Final answer: Domain: $\mathbb{R}$, Range: $[2, \infty)$.