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Section 1-5 : Functions of Several Variables

4. Find the domain of the following function.

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

There really isn’t all that much to this problem. We know that we can’t have division by zero and we can’t take square roots of negative numbers and so we’ll need to require that whatever \(\left( {x,y} \right)\) is it will need to satisfy the following three conditions.

\[x \ge - 1\hspace{0.5in}x \ne 0\hspace{0.5in}y \ge - 4\]

This is also our domain since these are the only conditions require in order for the function to exist.

A sketch of the domain is shown below. We can take any point in the green area or on the red lines with the exception of the \(y\)-axis (i.e. \(x \ne 0\)) as indicated by the black dashes on the \(y\)-axis.