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Magazine Chapter 13: Problem 42
Sketch the largest region on which the function \(f\) is continuous. $$ f(x, y)=\sqrt{x-y} $$
Short Answer
Expert verified
The region is \( \{ (x, y) \mid x \geq y \} \).
Step by step solution
01
Identify where the function is defined
The function \( f(x,y) = \sqrt{x-y} \) involves a square root. For the square root to be defined, its argument \( x-y \) must be greater than or equal to zero. Thus, we have the condition \( x-y \geq 0 \).
02
Simplify the inequality
From the inequality \( x-y \geq 0 \), we deduce that \( x \geq y \). This represents a half-plane in the coordinate plane, specifically the set of all points that lie on or above the line \( x = y \).
03
Determine the continuous region
Since the square root function is continuous wherever it is defined, \( f(x,y) = \sqrt{x-y} \) is continuous on the region including and above the line \( x = y \), which can be described as \( \{ (x, y) \mid x \geq y \} \).
04
Sketch the largest region
To sketch the region, draw the line \( x = y \) on the coordinate plane. The largest region where \( f \) is continuous includes all points above this line, and the line itself. This covers the entire half-plane where \( x-y \geq 0 \).
05
Conclusion
The largest region on which \( f \) is continuous is the set of points \( (x,y) \) such that \( x \geq y \). This region includes the boundary line \( x = y \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Root Function
The square root function is a basic mathematical operation that is commonly used in algebra. It involves finding a number which, when multiplied by itself, gives the original number. In equation form, the square root of a number \( a \) is written as \( \sqrt{a} \). The square root function is defined only for non-negative numbers. Here are some critical points to consider:
- For any non-negative number \( a \), \( \sqrt{a} \) is a real number.
- If \( a < 0 \), \( \sqrt{a} \) is not a real number.
- The function \( f(x) = \sqrt{x} \) is continuous and increasing for \( x \geq 0 \).
Inequalities
Inequalities are mathematical expressions involving ">", "<", "鈮�", or "鈮�". They signify a relationship between two expressions that are not equivalent. In the function \( f(x, y) = \sqrt{x-y} \), the inequality \( x-y \geq 0 \) plays a crucial role. Let's explore it:
- Understanding \( x-y \geq 0 \): This inequality implies that the value of \( x \) must be greater than or equal to \( y \). It dictates the set of points \((x, y)\) for which the function is defined and continuous.
- Simplifying inequalities: Inequalities can be manipulated like equations, but always consider the direction of the inequality sign. For example, from \( x-y \geq 0 \), we simplify to \( x \geq y \).
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can graph and analyze mathematical relationships and functions. It is divided into four quadrants by the x-axis and y-axis. Points on this plane are defined using pairs, \((x, y)\). For the function \( f(x, y) = \sqrt{x-y} \), the coordinate plane helps us understand where the function is defined and continuous:
- The line \( x = y \): This line is crucial as it serves as the boundary for continuity. Any point for which \( x \geq y \) belongs in our region of interest.
- Half-plane description: The inequality \( x \geq y \) describes a half-plane on the coordinate plane. It includes all points above and on the line \( x = y \).
- Graphical representation: To sketch this region, draw the line \( x = y \). The region where the function \( f(x, y) \) is both defined and continuous is the half-plane above and including this line.
