In Exercises 8 through 17, determine the region of...

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{x^{2}+y^{2}}{\sqrt{9-x^{2}-y^{2}}}

Key Concepts

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. For functions of several variables, the domain comprises those points (x, y) that satisfy all conditions imposed by the function’s operations, such as ensuring that expressions under radicals are nonnegative and denominators are nonzero.

Continuity of Functions

Continuity in multivariable functions means that the function does not have any abrupt jumps or breaks at the points within its domain. If every point in the domain has a limit equal to the function's value at that point, then the function is continuous over that domain.

Square Root Function Restrictions

The square root function requires that its argument be nonnegative to produce a real number. When a square root appears in a denominator, the argument must actually be strictly positive, ensuring the denominator is not zero and the function remains defined.

Inequalities in Domain Determination

Many functions involve conditions that lead to inequalities, which must be solved to determine the domain. For instance, setting the expression inside a square root greater than zero leads to an inequality that defines a region in the plane, often corresponding to familiar shapes like circles.

Graphical Representation of Domains

Visualizing the domain of a function in the plane involves drawing the region that satisfies the inequality or conditions identified. This graphical approach helps in understanding where the function is continuous by clearly showing the set of points over which the function is properly defined.

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