2.6 Finding Domain and Range (Techniques)

Domain: For square root to be defined, $9 - x^2 \ge 0$. $(3-x)(3+x) \ge 0$. Using wavy curve method: $x \in [-3, 3]$.

Range: Let $y = \sqrt{9 - x^2}$. Clearly $y \ge 0$ (square root is non-negative). Square both sides: $y^2 = 9 - x^2 \implies x^2 = 9 - y^2$. Since $x^2 \ge 0$, we have $9 - y^2 \ge 0 \implies y^2 \le 9 \implies -3 \le y \le 3$. Combining $y \ge 0$ and $y \in [-3, 3]$, we get Range = $[0, 3]$.