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Chapter 15: Problem 51

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$g(x, y)=\sqrt[3]{x^{2}+y^{2}-9}$$

Short Answer

Expert verified
Question: Determine the points in the real plane (R虏) where the given function $$g(x, y)=\sqrt[3]{x^{2}+y^{2}-9}$$ is continuous. Answer: The function is continuous for all points \((x, y) \in \mathbb{R}^{2}\) such that \((x^2+y^2 \geq 9)\).

Step by step solution

01

Determine the Domain of the Function

First, let's determine the domain of the function \(g(x,y)\). The function is defined for all \((x,y)\) that satisfy the condition: $$x^{2}+y^{2}-9 \geq 0$$
02

Solve the Inequality

Now let's analyze the inequality \(x^{2}+y^{2}-9 \geq 0\): $$x^2+y^2 \geq 9$$ This inequality describes a closed circle with center at the origin and radius 3, including the points on the circle as well.
03

Determine the Continuity

The function \(g(x, y)=\sqrt[3]{x^{2}+y^{2}-9}\) is defined for all \((x,y)\) that satisfy the inequality \(x^2+y^2 \geq 9\). Therefore, the function is continuous for all points inside the closed circle \((x^2 + y^2 \geq 9)\), including the points on the circle. So the function is continuous for all points \((x, y) \in \mathbb{R}^{2}\) such that \((x^2+y^2 \geq 9)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of a Function
When tackling a function, especially one involving two variables, understanding its domain is crucial. The domain of a function refers to all the possible input values, essentially telling us where the function is defined. For a function \(g(x, y)\), we explore what conditions \((x, y)\) need to satisfy for \(g\) to have real outputs.

Take, for instance, the function \(g(x, y) = \sqrt[3]{x^2 + y^2 - 9}\). The cube root function itself is defined across all real numbers, meaning it's quite flexible. However, the issue lies in the expression under the cube root. Here it鈥檚 \(x^2 + y^2 - 9\), which must be non-negative. Thus, we solve the inequality \(x^2 + y^2 \geq 9\) to find valid \((x, y)\) pairs.

This inequality represents a circle centered at the origin with a radius of 3, including its boundary. Therefore, every point \((x, y)\) on or outside the boundary of this circle forms the domain of our function.
Function of Two Variables
Functions involving two variables, like our function \(g(x, y)\), have input in the form of pairs \((x, y)\). This is a familiar concept if you think about geographical coordinates: an x-value represents east/west, while a y-value stands for north/south.

Functions of two variables extend into three-dimensional space. The output is often visualized as a surface stretching above its specified domain on a grid. With \(g(x, y) = \sqrt[3]{x^2 + y^2 - 9}\), imagine the rgbotloveraid through a circle; points outside that circle fulfill the function's requirements for input.

Ultimately, understanding functions of two variables opens the door to studying relationships across dimensions鈥攁n essential tool in geometry, physics, and beyond.
Continuous Functions
A continuous function is one where small changes in input produce small changes in the output, lacking any breaks, jumps, or holes. For functions of two variables, examining continuity involves checking every potential path approaching a point on the function鈥檚 surface.

The function \(g(x, y) = \sqrt[3]{x^2 + y^2 - 9}\) is continuous within its domain. Since cube root functions are naturally continuous across all real numbers, once our domain stipulation \(x^2 + y^2 \geq 9\) is met, there is continuity across this defined region.

Understanding a function's continuity helps in predicting and managing transitions smoothly鈥攊mperative for applications like computer graphics and mathematical modeling, where sudden shifts would be problematic.

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Most popular questions from this chapter

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