domain using level curves?

Question

Let $f(x,y)=4x^2-y^2$. The problem is to determine the range of this function using the idea of level curves. So one sets $f(x,y)=D$, where $D\in\mathbb{R}$. What values can $D$ take?

Answer 1

The range is $\mathbb{R}$

• if $D<0$ then $f(0,\sqrt{|D|})=-|D|=D$
• if $D>0$ then $f(1/2\sqrt{D})=D$
• if $D=0$ take $f(0,0)=0$