1. **Problem:** Find the natural domain of the function $f(x) = \sqrt{x^2 - 5x + 6}$. 2. **Formula and rules:** The natural domain of a square root function requires the radicand (expression inside the root) to be non-negative: $$x^2 - 5x + 6 \geq 0$$ 3. **Intermediate work:** Factor the quadratic: $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ 4. **Solve inequality:** $$(x - 2)(x - 3) \geq 0$$ 5. **Test intervals:** The critical points split the number line into intervals: - $(-\infty, 2)$ - $(2, 3)$ - $(3, \infty)$ 6. **Sign analysis:** - For $x < 2$, both $(x-2)$ and $(x-3)$ are negative, product is positive. - For $2 < x < 3$, $(x-2)$ positive, $(x-3)$ negative, product negative. - For $x > 3$, both positive, product positive. 7. **Domain conclusion:** $$(-\infty, 2] \cup [3, \infty)$$ **Final answer:** The natural domain of $f(x)$ is $$\boxed{(-\infty, 2] \cup [3, \infty)}$$