1. We are given four functions and asked to find and sketch their domain, range, level curves, and region properties. 2. Let's analyze each function one by one. (i) $f(x,y) = \sqrt{9 - x^2 - y^2}$ - Domain: The expression inside the square root must be non-negative: $9 - x^2 - y^2 \geq 0$ which implies $x^2 + y^2 \leq 9$. - Range: Since the square root outputs non-negative values, the range is $[0,3]$. - Level curves: For a constant $c$, $f(x,y) = c$ means $\sqrt{9 - x^2 - y^2} = c$ or $9 - x^2 - y^2 = c^2$ which rearranges to $x^2 + y^2 = 9 - c^2$. - These are circles centered at the origin with radius $\sqrt{9 - c^2}$ for $0 \leq c \leq 3$. (ii) $f(x,y) = \ln(x^2 + y^2 - 4)$ - Domain: The argument of the logarithm must be positive: $x^2 + y^2 - 4 > 0$ which implies $x^2 + y^2 > 4$. - Range: Since $\ln$ can take any real value, the range is $(-\infty, \infty)$. - Level curves: For constant $c$, $\ln(x^2 + y^2 - 4) = c$ implies $x^2 + y^2 - 4 = e^c$ or $x^2 + y^2 = 4 + e^c$. - These are circles centered at the origin with radius $\sqrt{4 + e^c}$. (iii) $f(x,y) = \frac{1}{x^2 + y^2}$ - Domain: $x^2 + y^2 \neq 0$ to avoid division by zero, so domain is all points except $(0,0)$. - Range: Since $x^2 + y^2 > 0$, $f(x,y)$ is positive and can be arbitrarily large near $(0,0)$ and approaches 0 as $x^2 + y^2 \to \infty$, so range is $(0, \infty)$. - Level curves: For constant $c$, $\frac{1}{x^2 + y^2} = c$ implies $x^2 + y^2 = \frac{1}{c}$. - These are circles centered at the origin with radius $\sqrt{\frac{1}{c}}$. (iv) $f(x,y) = \sqrt{y - x - 2}$ - Domain: The expression inside the square root must be non-negative: $y - x - 2 \geq 0$ which implies $y \geq x + 2$. - Range: Since square root outputs non-negative values, range is $[0, \infty)$. - Level curves: For constant $c$, $\sqrt{y - x - 2} = c$ implies $y - x - 2 = c^2$ or $y = x + 2 + c^2$. - These are straight lines with slope 1 and varying intercepts. Summary: - Domains are defined by inequalities ensuring the function expressions are valid. - Ranges depend on the function type (square root, logarithm, reciprocal). - Level curves are circles or lines depending on the function. "slug":"domain range level curves","subject":"multivariable calculus","desmos":{"latex":"","features":{"intercepts":true,"extrema":true}},"q_count":4