Question

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

Multivariable functions
ANSWERED
asked 2020-10-28
What tis the complete domain D and range R of the following multivariable functions: \(\displaystyle{w}{\left({x},{y}\right)}=\sqrt{{{y}-{4}{x}^{{2}}}}\)

Answers (1)

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 \(\displaystyle{w}{\left({x},{y}\right)}\ge{0}\)
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