1. The problem asks to find and sketch the domain of each function given. 2. The domain of a function $f(x,y)$ is the set of all points $(x,y)$ for which the function is defined. 3. For each function, analyze the expression inside to determine restrictions such as division by zero or taking logarithm of non-positive numbers. (i) $f(x,y) = 9 - x^2 - y^2$ - This is a polynomial expression, defined for all real $x,y$. - Domain: $\{(x,y) \in \mathbb{R}^2\}$ (all real plane). (ii) $f(x,y) = \ln(x^2 + y^2 - 4)$ - Logarithm requires argument $>0$. - So, $x^2 + y^2 - 4 > 0 \Rightarrow x^2 + y^2 > 4$. - Domain: outside the circle radius 2, $\{(x,y) : x^2 + y^2 > 4\}$. (iii) $f(x,y) = \frac{1}{x^2 + y^2}$ - Denominator cannot be zero. - So, $x^2 + y^2 \neq 0 \Rightarrow (x,y) \neq (0,0)$. - Domain: all points except origin. (iv) $f(x,y) = \sqrt{y - x - 2}$ - Square root requires argument $\geq 0$. - So, $y - x - 2 \geq 0 \Rightarrow y \geq x + 2$. - Domain: region on or above the line $y = x + 2$. Final answers: (i) Domain: $\mathbb{R}^2$ (ii) Domain: $\{(x,y) : x^2 + y^2 > 4\}$ (iii) Domain: $\mathbb{R}^2 \setminus \{(0,0)\}$ (iv) Domain: $\{(x,y) : y \geq x + 2\}$