Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = ( - 4 x - 3 y ) ^ { 3 }$$
A) Domain: all points in the $x y$-plane; range: real numbers $z \leq 0$; level curves: lines $- 4 x - 3 y = c$, $c \leq 0$
B) Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $- 4 x - 3 y = c$, $\geq \geq 0$
C) Domain: all points in the $x y$-plane; range: real numbers $z \geq 0$; level curves: lines $- 4 x - 3 y = c$
D) Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $- 4 x - 3 y = c$

Question

Quizplus · Accepted Answer

The function $f(x, y) = (-4x - 3y)^3$ can take any pair of real numbers as inputs, so its domain is all points in the $xy$-plane. The cubic function can produce all real numbers as outputs, so the range is all real numbers. The level curves are defined by setting $f(x, y)$ equal to a constant, which means $(-4x - 3y)^3 = c$, and taking the cube root of both sides gives $-4x - 3y = c^{1/3}$, which represents lines in the $xy$-plane.

Find the Domain and Range and Describe the Level Curves f(x,y)=(−4x−3y)3f ( x , y ) = ( - 4 x - 3 y ) ^ { 3 }f(x,y)=(−4x−3y)3

Find the Domain and Range and Describe the Level Curves $f ( x , y ) = ( - 4 x - 3 y ) ^ { 3 }$