Problem 22 - Calculus for Biology and Medicine

Question

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=\frac{x}{y} ; D=\\{(x, y): 0 \leq x \leq 1,1 \leq y \leq 2\\}\)

Step-by-Step Solution

Answer

The range of the function is \([0, 1]\).

Step 1: Identify the Function and Domain

The function is given as \(f(x, y) = \frac{x}{y}\) and the domain is \(D = \{(x, y) : 0 \leq x \leq 1, 1 \leq y \leq 2\}\). This means \(x\) can vary between 0 and 1, and \(y\) can vary between 1 and 2.

Step 2: Examine the Function Behavior

Since \(f(x, y) = \frac{x}{y}\), the function depends on how both \(x\) and \(y\) vary. The maximum value for \(\frac{x}{y}\) occurs when \(x\) is maximized and \(y\) is minimized. Conversely, the minimum value occurs when \(x\) is minimized and \(y\) is maximized.

Step 3: Determine Maximum Value

The function \(f\) reaches its maximum when \(x\) is maximum, i.e., 1, and \(y\) is minimum, i.e., 1. Therefore, the maximum value is \(f(1, 1) = \frac{1}{1} = 1\).

Step 4: Determine Minimum Value

The function \(f\) reaches its minimum when \(x\) is minimum, i.e., 0, and \(y\) is maximum, i.e., 2. Therefore, the minimum value is \(f(0, 2) = \frac{0}{2} = 0\).

Step 5: Conclude the Range

Since the function increases as \(x\) increases and decreases as \(y\) increases, with the calculations above, any intermediate values of \(x\) and \(y\) will yield values of \(f(x, y)\) between 0 and 1. Thus, the range of \(f(x, y)\) is \([0, 1]\).

Key Concepts

Function Range

Understanding the range of a function is crucial in mathematical analysis. It tells us all possible values a function can take when applied to its domain. For the function \(f(x, y) = \frac{x}{y}\), we consider its behavior over the specified domain \(D = \{(x, y) : 0 \leq x \leq 1, 1 \leq y \leq 2\}\).

• The maximum value of \(f\) occurs when \(x\) is 1 and \(y\) is 1, giving us \(f(1, 1) = 1\).
• The minimum value of \(f\) is when \(x\) is 0 and \(y\) is 2, resulting in \(f(0, 2) = 0\).

Hence, the range of the function, or all possible outputs, lies between these two extremes: - From 0 to 1.
This means every possible value produced by \(f(x, y)\) as \((x,y)\) varies within the domain belongs to the closed interval \([0, 1]\). Understanding this concept helps visualize the implication of the function on its domain.

Two-Variable Functions

Two-variable functions, like \(f(x, y) = \frac{x}{y}\), depend on two independent variables: \(x\) and \(y\). Unlike single-variable functions, they map ordered pairs of numbers into a single real number.
Such functions allow us to examine interactions between two changing parameters. For example, when both \(x\) and \(y\) vary, the value of \(f(x, y)\) reveals how variations in one variable affect the entire expression. Some useful properties of two-variable functions include:

• They exhibit different behaviors depending on domain restrictions - for this function, \(x\) influences the numerator while \(y\) dictates the denominator.
• Their visualization often involves surfaces in three-dimensional space, providing insights into how different pairs \((x, y)\) yield distinct outcomes.

These functions underpin more complex topics in Calculus, such as partial derivatives and surface integrals, making them vital in technical fields and studies.

Domain Analysis

Analyzing a function's domain is a fundamental step. It helps in understanding where the function is defined and reveals applicable constraints. The domain of a function is essentially all ordered pairs \((x, y)\) for which the function produces a valid result. For \(f(x, y) = \frac{x}{y}\), the domain ensures that we do not have division by zero, so \(y\) must never be zero. In the specified domain \(D = \{(x, y): 0 \leq x \leq 1, 1 \leq y \leq 2\}\), \(y\) comfortably stays away from zero, ensuring valid outputs. Key points about domain analysis include:

• Identify constraints - here, \(x\) and \(y\)'s ranges enforce valid expressions throughout the specified domain.
• Recognize that each valid input \((x, y)\) leads to a unique value of \(f(x, y)\), here ranging from 0 to 1.

A thorough domain analysis prevents errors and provides a clear picture of how a function behaves within its boundary conditions, aiding in effective application and exploration of mathematical models.