Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = \frac { 1 } { 4 x ^ { 2 } + 5 y ^ { 2 } }$$
A) Domain: all points in the $x y$-plane except $( 0,0 )$; range: all real numbers; level curves; ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$
B) Domain: all points in the xy-plane except $( 0,0 )$; range: real numbers $ 0$; level curves: ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$
C) Domain: all points in the $x y$-plane; range: all real numbers; level curves; ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$
D) Domain: all points in the $x y$-plane; range: real numbers $ 0$; level curves: ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$

Question

Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = \frac { 1 } { 4 x ^ { 2 } + 5 y ^ { 2 } }$$
A) Domain: all points in the $x y$-plane except $( 0,0 )$; range: all real numbers; level curves; ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$
B) Domain: all points in the xy-plane except $( 0,0 )$; range: real numbers $> 0$; level curves: ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$
C) Domain: all points in the $x y$-plane; range: all real numbers; level curves; ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$
D) Domain: all points in the $x y$-plane; range: real numbers $> 0$; level curves: ellipses $4 x ^ { 2 } + 5 y ^ { 2 } = c$

Quizplus · Accepted Answer

The denominator of the function can never be zero, so the only point that is not in the domain is (0,0). Therefore, the domain is all points in the xy-plane except (0,0).
Since the denominator is always positive, the range is all positive real numbers.
The level curves are given by:
\begin{align*}
\frac{1}{4x^2 + 5y^2} &= c \
4x^2 + 5y^2 &= \frac{1}{c} \
\end{align*}
This is the equation of an ellipse centered at the origin with semi-axes lengths of $\sqrt{\frac{1}{4c}}$ and $\sqrt{\frac{1}{5c}}$. As $c$ increases, the ellipse gets smaller. Therefore, the level curves are ellipses $4x^2 + 5y^2 = c$ with $c > 0$. This means that choice B is the only correct choice.

Find the Domain and Range and Describe the Level Curves f(x,y)=14x2+5y2f ( x , y ) = \frac { 1 } { 4 x ^ { 2 } + 5 y ^ { 2 } }f(x,y)=4x2+5y21​

Find the Domain and Range and Describe the Level Curves $f ( x , y ) = \frac { 1 } { 4 x ^ { 2 } + 5 y ^ { 2 } }$