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Chapter 12: Problem 13

Find the domain of the following functions. $$f(x, y)=\sqrt{25-x^{2}-y^{2}}$$

Short Answer

Expert verified
Answer: The domain of the function $$f(x, y)$$ is the set of all points (x, y) that lie inside or on the circle with radius 5 and centered at the origin. In set notation, the domain is represented as $$\{(x,y)\ |\ x^{2}+y^{2}\leq 25\}$$.

Step by step solution

01

Write down the inequality

To find the domain, we need to ensure the expression inside the square root is greater than or equal to 0, because taking the square root of negative numbers would result in complex values. So, we need to solve the inequality: $$25-x^{2}-y^{2}\geq0$$
02

Rearrange the inequality

Rearrange the equation so that all the terms are on one side: $$x^{2}+y^{2}\leq 25$$
03

Interpret the inequality

The inequality $$x^{2}+y^{2}\leq 25$$ represents all points (x, y) for which the sum of their squared values is less than or equal to 25. Geometrically, this represents a closed circle centered at the origin (0, 0) and with a radius 5, because the equation $$x^{2}+y^{2}=25$$ is the equation of a circle with radius $$\sqrt{25}=5$$.
04

Write down the domain

Now that we have interpreted the inequality, we can state the domain. The domain of the function $$f(x,y)$$ includes all the points (x, y) that lie inside or on the circle with radius 5 and centered at the origin. In set notation, the domain of $$f(x, y)$$ is as follows: $$\{(x,y)\ |\ x^{2}+y^{2}\leq 25\}$$

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