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Section 1-5 : Functions of Several Variables

3. Find the domain of the following function.

\[f\left( {x,y,z} \right) = \frac{1}{{{x^2} + {y^2} + 4z}}\] Show Solution

There really isn’t all that much to this problem. We know that we can’t have division by zero so we’ll need to require that whatever \(\left( {x,y,z} \right)\) is it will need to satisfy,

\[{x^2} + {y^2} + 4z \ne 0\]

Since this is the only condition we need to meet this is also the domain of the function.

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

\[4z \ne - {x^2} - {y^2}\hspace{0.5in} \Rightarrow \hspace{0.5in}z \ne - \frac{{{x^2}}}{4} - \frac{{{y^2}}}{4}\]

So, it looks like we need to avoid points, \(\left( {x,y,z} \right)\), that are on the elliptic paraboloid given by the equation above.