1. **Problem statement:** Find the domain of the functions: (a) $f(x,y) = \sqrt{25 - x^2 - 4y^2}$ (b) $g(x,y,z) = \ln(x + y^2 + z - 2)$ --- 2. **Domain of (a):** The square root function requires the radicand to be non-negative: $$25 - x^2 - 4y^2 \geq 0$$ Rearranging: $$x^2 + 4y^2 \leq 25$$ This inequality describes an ellipse centered at the origin with semi-major and semi-minor axes. - The domain is all points $(x,y)$ inside or on the ellipse defined by $x^2 + 4y^2 \leq 25$. --- 3. **Domain of (b):** The natural logarithm function requires its argument to be strictly positive: $$x + y^2 + z - 2 > 0$$ Rearranging: $$x + y^2 + z > 2$$ This inequality defines a region in 3D space where the sum of $x$, $y^2$, and $z$ is greater than 2. - The domain is all points $(x,y,z)$ such that $x + y^2 + z > 2$. --- **Final answers:** - Domain of $f$: $\{(x,y) \mid x^2 + 4y^2 \leq 25\}$ - Domain of $g$: $\{(x,y,z) \mid x + y^2 + z > 2\}$