1. **State the problem:** Find the domain and range of the function $$r(x) = \sqrt{4 - x^2 + 1}$$. 2. **Simplify the function:** $$r(x) = \sqrt{5 - x^2}$$ 3. **Find the domain:** The expression inside the square root must be non-negative: $$5 - x^2 \geq 0$$ 4. **Solve the inequality:** $$x^2 \leq 5$$ $$-\sqrt{5} \leq x \leq \sqrt{5}$$ 5. **Domain:** $$\boxed{[-\sqrt{5}, \sqrt{5}]}$$ 6. **Find the range:** Since the square root outputs non-negative values, the minimum value is 0 when the inside is 0: $$5 - x^2 = 0 \Rightarrow x = \pm \sqrt{5}$$ The maximum value is when $$x=0$$: $$r(0) = \sqrt{5 - 0} = \sqrt{5}$$ 7. **Range:** $$\boxed{[0, \sqrt{5}]}$$