1. Find the domain and range of each function. 2. For graphing functions 1 to 12: - 1. $f(x,y) = y - x$ - Domain: All real numbers $x, y \in \mathbb{R}$. - Range: All real numbers $\mathbb{R}$ since $y - x$ can be any real number. - 2. $f(x,y) = \sqrt{y} - x$ - Domain: $y \geq 0$, $x \in \mathbb{R}$. - Range: All real numbers because $\sqrt{y} \geq 0$ and $-x$ shifts it freely. - 3. $f(x,y) = 4x^2 + 9y^2$ - Domain: All real $x,y$. - Range: $f(x,y) \geq 0$, minimum 0 at $(0,0)$. - 4. $f(x,y) = x^2 - y^2$ - Domain: All real $x,y$. - Range: All real numbers (difference of squares can yield any value). - 5. $f(x,y) = \frac{1}{\sqrt{16 - x^2 - y^2}}$ - Domain: $16 - x^2 - y^2 > 0$ or $x^2 + y^2 < 16$, the interior of a circle radius 4. - Range: $(\frac{1}{4}, \infty)$ since denominator approaches 0 near boundary. - 6. $f(x,y) = xy$ - Domain: All real $x,y$. - Range: All real numbers. - 7. $f(x,y) = \frac{y}{x^2}$ - Domain: $x \neq 0$, all $y$. - Range: All real numbers. - 8. $f(x,y) = \ln(x^2 + y^2)$ - Domain: $x^2 + y^2 > 0$ (all except origin). - Range: $(-\infty, \infty)$. - 9. $f(x,y) = e^{-(x^2 + y^2)}$ - Domain: All real $x,y$. - Range: $(0,1]$, maximum 1 at $(0,0)$. - 10. $f(x,y) = \sin^{-1}(y - x)$ - Domain: $-1 \leq y - x \leq 1$. - Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$. - 11. $f(x,y) = \tan^{-1}(\frac{y}{x})$ - Domain: $x \neq 0$. - Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$. - 12. $f(x,y) = \sqrt{9 - x^2 - y^2}$ - Domain: $x^2 + y^2 \leq 9$. - Range: $[0,3]$. Summary of key plotting features: - Most domains are typical like entire $\mathbb{R}^2$ or constrained by the square root or denominator. - Ranges vary from all real numbers to bounded intervals. Since graph plotting is requested for 1 to 12, one can plot each function in 3D to see shape: - E.g., for $f(x,y) = y - x$ plane slanting diagonally. - Elliptic paraboloid for $4x^2 + 9y^2$. - Hyperbolic paraboloid like in $x^2 - y^2$. - Circular domains for root functions.