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Chapter 12: Problem 14

Find the domain of the following functions. $$f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}-25}}$$

Short Answer

Expert verified
Answer: The domain of the function is all points (x, y) in the plane such that they are strictly outside the circle of radius 5 centered at the origin, which can be represented by the inequality $$x^{2} + y^{2} > 25$$.

Step by step solution

01

Identify the restrictions

In this function, the restrictions are in the denominator. We have a square root, so the expression inside the square root must be non-negative, and we cannot have a denominator of zero, so the expression inside the square root cannot be zero either.
02

Set up the inequality

To ensure the denominator is non-zero and the expression inside the square root is non-negative, we will set up the following inequality: $$x^{2} + y^{2} - 25 > 0$$
03

Rearrange the inequality

Rearrange the inequality to a more recognizable form. Adding 25 to both sides, we get: $$x^{2} + y^{2} > 25$$
04

Identify the domain

We now have the inequality that defines the domain of the function. The inequality relates to the equation of a circle: $$x^{2} + y^{2} = 25$$ This is the equation of a circle with a radius of 5 centered at the origin. The inequality states that the function is only defined for (x, y) points outside this circle (excluding the boundary): $$x^{2} + y^{2} > 25$$ So the domain of the function is all points (x, y) in the plane such that they are strictly outside the circle of radius 5 centered at the origin.

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