Question

# What tis the complete domain D and range R of the following multivariable functions: w(x,y)=sqrt(y-4x^2)

Multivariable functions
What tis the complete domain D and range R of the following multivariable functions: $$\displaystyle{w}{\left({x},{y}\right)}=\sqrt{{{y}-{4}{x}^{{2}}}}$$

2020-10-29
$$\displaystyle{w}{\left({x},{y}\right)}=\sqrt{{{y}-{4}{x}^{{2}}}}$$
$$\displaystyle{y}−{4}{x}^{{2}}\ge{0}$$
$$\displaystyle−{4}{x}^{{2}}\ge−{y}$$
$$\displaystyle−{1}\times{\left(−{4}{x}^{{2}}\right)}\le−{1}×{\left(−{y}\right)}$$
$$\displaystyle{4}{x}^{{2}}\le{y}$$
$$\displaystyle{x}^{{2}}{>}{0}$$
$$\displaystyle\Rightarrow{4}{x}^{{2}}{>}{0}$$
$$\displaystyle\Rightarrow{y}\ge{0}$$
$$\displaystyle{D}=\mathbb{R}^{{2}}$$ such that $$\displaystyle{y}\ge{0}{\quad\text{and}\quad}{4}{x}^{{2}}\le{y}.$$
$$\displaystyle\sqrt{{{y}-{4}{x}^{{2}}}}\ge{0}$$
So, $$\displaystyle{w}\ge{0}$$
$$\displaystyle{R}\in\mathbb{R}$$ such $$\displaystyle{w}{\left({x},{y}\right)}\ge{0}$$