Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = 4 x ^ { 2 } - 2 y ^ { 2 }$$
A) Domain: all points in the first quadrant of the $x y$-plane; range: all real numbers; level curves: hyperbolas $4 x ^ { 2 } - 2 y ^ { 2 } = c$
B) Domain: all points in the $x y$-plane; range: all real numbers; level curves: hyperbolas $4 x ^ { 2 } - 2 y ^ { 2 } = c$
C) Domain: all points in the $x y$-plane; range: real numbers $\mathrm { z } \geq 0$; level curves: ellipses $4 x ^ { 2 } + 2 y ^ { 2 } = c$
D) Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses $4 x ^ { 2 } + 2 y ^ { 2 } = c$

Question

Quizplus · Accepted Answer

The function $f(x, y) = 4x^2 - 2y^2$ is defined for all real numbers $x$ and $y$, so its domain is all points in the $xy$-plane. The function can take any real number value, making its range all real numbers. The level curves are given by setting $f(x, y) = c$, which results in hyperbolas $4x^2 - 2y^2 = c$.

Find the Domain and Range and Describe the Level Curves f(x,y)=4x2−2y2f ( x , y ) = 4 x ^ { 2 } - 2 y ^ { 2 }f(x,y)=4x2−2y2

Find the Domain and Range and Describe the Level Curves $f ( x , y ) = 4 x ^ { 2 } - 2 y ^ { 2 }$