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Chapter 10: Problem 23

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=\frac{x}{y} ; D=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1, y>x]\)

Short Answer

Expert verified
The range of the function is \((0, 1)\).

Step by step solution

01

Understanding the Function and Domain

The function given is \( f(x, y) = \frac{x}{y} \). We need to find the range of this function, which means finding all possible values \( f(x, y) \) can take, given the domain constraints: \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 1 \), and \( y > x \).
02

Analyzing the Domain Constraints

The domain constraint \( y > x \) implies that for any point \((x, y)\) in \( D \), \( y \) must be strictly greater than \( x \). This constraint means that \( x = y \) is not allowed within the domain, so the function is defined as long as the numerator \( x \) is less than the denominator \( y \).
03

Expressing the Function Value

The function value is \( f(x, y) = \frac{x}{y} \). Since \( x < y \), this implies that \( \frac{x}{y} < 1 \). However, because \( x \) can range very close to \( y \) (but not reach it because \( y > x \)), \( \frac{x}{y} \) can become very close to 1. Also, as \( x \rightarrow 0 \), \( \frac{x}{y} \rightarrow 0 \).
04

Determining Range Bounds

Given \( 0 \leq x < y \leq 1 \), the smallest value \( \frac{x}{y} \) can approach is 0 (when \( x = 0 \)), and it can approach 1 as \( x \) becomes close to \( y \). Thus, the range of \( f(x, y) \) is all values in the interval \( (0, 1) \), meaning 0 is not included, but all values up to but not including 1 are present.

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Key Concepts

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

Function Range
Understanding the range of a function is a crucial aspect of multivariable calculus. It involves determining all possible output values of a function given certain constraints. In this particular exercise, we analyze the range of the function \( f(x, y) = \frac{x}{y} \) within the domain \( D \). The range is determined based on the behavior of the function as its variables \( x \) and \( y \) vary within their constraints.
  • The function \( \frac{x}{y} \) takes input from the domain where \( 0 \leq x < y \leq 1 \).
  • When \( x \) approaches zero, \( \frac{x}{y} \) becomes closer to 0, showing it can take values near zero.
  • Conversely, as \( x \) approaches \( y \), the fraction approaches 1 without ever reaching it due to the condition \( y > x \).
Thus, the range of the function is the interval \( (0, 1) \), meaning it can yield any value greater than 0 and less than 1 as \( x \) and \( y \) satisfy their respective constraints.
Domain Constraints
Domain constraints define the permissible set of input values for a function, acting as limits within which the function can operate. For the function \( f(x, y) = \frac{x}{y} \), the exercise specifies a domain \( D = \{(x, y): 0 \leq x \leq 1, 0 \leq y \leq 1, y > x\} \). These constraints are vital in shaping the behavior and range of the function.
  • The first two constraints, \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \), specify that both \( x \) and \( y \) must lie within the unit interval.
  • The additional constraint \( y > x \) excludes scenarios where \( y \) equals or is less than \( x \), ensuring the validity of the function \( \frac{x}{y} \) by preventing division by zero and infinite values.
This setup means the function is defined in a triangular region in the \( xy \)-plane, a specific portion of the unit square, which importantly influences the possible output values or the range of the function.
Two-Variable Functions
Two-variable functions, such as \( f(x, y) = \frac{x}{y} \), are functions that depend on more than one independent variable. This introduces a whole new layer of complexity compared to single-variable functions, where we only consider one input.
  • With two-variable functions, we need to understand how changes in one variable affect the function's output while keeping the other constant.
  • Visualizing such functions often involves considering three-dimensional surfaces or topography where each coordinate pair \((x, y)\) maps to a value of the function.
In the given context, the function \( \frac{x}{y} \) shows how the relationship and constraints between \( x \) and \( y \) affect possible function values. Understanding two-variable functions is essential when solving problems in physics, engineering, and economics, where multiple inputs commonly interact to produce outcomes.

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Most popular questions from this chapter

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(-1.1)}\left(2 x y+y^{2}\right)\)

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