1. **State the problem:** We are given the function $h(x) = \sqrt{4x + 12}$ and we want to understand its properties and possibly simplify or analyze it. 2. **Recall the formula and rules:** 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. **Determine the domain:** Since $h(x) = \sqrt{4x + 12}$, the expression inside the square root must be non-negative: $$4x + 12 \geq 0$$ 4. **Solve the inequality:** $$4x \geq -12$$ $$x \geq -3$$ So the domain of $h(x)$ is all real numbers $x$ such that $x \geq -3$. 5. **Simplify the function if possible:** $$h(x) = \sqrt{4x + 12} = \sqrt{4(x + 3)} = \sqrt{4} \cdot \sqrt{x + 3} = 2\sqrt{x + 3}$$ 6. **Interpretation:** The function $h(x)$ is twice the square root of $x + 3$, defined for $x \geq -3$. 7. **Summary:** - Domain: $x \geq -3$ - Simplified form: $h(x) = 2\sqrt{x + 3}$ This function outputs real numbers for all $x$ in its domain and grows as $x$ increases.