1. The problem is to analyze the function $h(x) = \sqrt{x-4}$. 2. The square root function $\sqrt{y}$ is defined only for $y \geq 0$ because the square root of a negative number is not a real number. 3. Therefore, for $h(x)$ to be defined, the expression inside the square root must be non-negative: $$x - 4 \geq 0$$ 4. Solve the inequality: $$x \geq 4$$ 5. This means the domain of $h(x)$ is all real numbers $x$ such that $x \geq 4$. 6. The function $h(x)$ outputs the positive square root of $x-4$, so it is increasing for $x \geq 4$. 7. The graph of $h(x)$ starts at the point $(4,0)$ and increases as $x$ increases. Final answer: The domain of $h(x) = \sqrt{x-4}$ is $[4, \infty)$ and the function is defined and real-valued only for $x \geq 4$.