Find the domain of each function and graph the domain. Use a...

Find the domain of each function and graph the domain. Use a solid curve to indicate that the domain includes the boundary and a dashed curve to indicate that the domain excludes the boundary.
$$
f(x, y)=\frac{5}{\sqrt{2 y+x^{2}}}
$$

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (for functions of several variables, points in space) for which the function is defined. When determining the domain, it's essential to identify any restrictions imposed by operations in the function, such as division by zero or taking square roots of negative numbers.

Restrictions Due to Square Roots and Denominators

Functions that involve square roots require the radicand to be non-negative in order to produce real-number outputs. In cases where the square root is in the denominator, as in this function, the radicand must be strictly positive to avoid division by zero. This leads to an inequality that must be satisfied, defining the domain of the function.

Inequalities in Multivariable Functions

When dealing with functions of two variables, inequalities like 2y + x^2 > 0 arise from the conditions necessary for the function to be defined. Solving these inequalities involves understanding how the variables interact, resulting in a region in the plane that represents the allowable input values.

Graphical Representation of Domains

Graphing the domain of a function involves depicting the region determined by the inequality on the coordinate plane. Boundaries are represented using curves; a solid curve indicates that the boundary is included in the domain, while a dashed curve shows that the boundary is excluded. In this problem, since the inequality is strict (i.e., > rather than ?), any boundary defined by the equation 2y + x^2 = 0 is depicted with a dashed curve to show that these points are not part of the domain.

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