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### Section 1-5 : Functions of Several Variables

1. Find the domain of the following function.

$f\left( {x,y} \right) = \sqrt {{x^2} - 2y}$ Show Solution

There really isn’t all that much to this problem. We know that we can’t have negative numbers under the square root and so the we’ll need to require that whatever $$\left( {x,y} \right)$$ is it will need to satisfy,

${x^2} - 2y \ge 0$

Let’s do a little rewriting on this so we can attempt to sketch the domain.

${x^2} \ge 2y\hspace{0.5in} \Rightarrow \hspace{0.5in}y \le \frac{1}{2}{x^2}$

So, it looks like we need to be on or below the parabola above. The domain is illustrated by the green area and red line in the sketch below. 