Problem 10 - Calculus and its applications

Question

Determine the domain of each function of two variables. $$ g(x, y)=\ln \left(x^{2}-y\right) $$

Step-by-Step Solution

Answer

The domain is all \((x, y)\) such that \( y < x^2 \).

Step 1: Understanding the Function

We need to find the domain of the function \( g(x, y) = \ln(x^2 - y) \). The domain comprises all \((x, y)\) pairs that make the expression inside the logarithm valid and greater than zero, since the logarithm of zero or a negative number is undefined.

Step 2: Setting Up Inequality

The expression inside the logarithm is \( x^2 - y \). To ensure it is valid for the logarithm, set up the inequality as \( x^2 - y > 0 \). This means \( y < x^2 \), which gives us the condition for the domain of the function.

Step 3: Describing the Domain

The domain of the function \( g(x, y) = \ln(x^2 - y) \) is the set of all \((x, y)\) such that \( y < x^2 \). In other words, \((x, y)\) must lie below the parabola \( y = x^2 \) in the coordinate plane.

Key Concepts

Domain of a Function

In mathematics, the domain of a function refers to the set of all possible input values (often represented as \(x\)) that a function can accept without resulting in any undefined or invalid operations. Understanding the domain is crucial because it determines where a function is applicable.
In simple terms, think of the domain as the limitations or boundaries within which a function "lives." Anything outside that boundary results in an incorrect or undefined output.

• For single-variable functions like \(f(x) = \text{ln}(x-5)\), the domain is determined by ensuring expressions inside functions like logarithms remain positive.
• For functions of two variables, such as \(g(x, y) = \text{ln}(x^2 - y)\), the domain becomes a set of \(x, y\) pairs that satisfy the condition necessary for the function to be defined.

Finding the domain of a complex function ensures that you understand the function's restrictions and how it behaves within its applicable realm.

Functions of Two Variables

Functions of two variables are a natural extension of single-variable functions into the multidimensional space. Instead of depending on a single input, these functions rely on two different inputs to determine their output, often represented as \((x, y)\).
This is especially useful in real-world scenarios where outcome depends on two variables, like in economics or the study of movement over a surface.

• For example, the function \(z = f(x, y)\) represents a surface in a 3D space where \(z\) depends on both \(x\) and \(y\).
• Visualizing these surfaces can help to understand how the output \(z\) changes as the two variables \(x\) and \(y\) change.

Understanding functions of two variables involves identifying and analyzing how changes in one or both variables affect the overall function. Key aspects include:

• The domain: identifying all possible \((x, y)\) combinations that will keep the function defined.
• The range: understanding the possible outputs from the function based on its domain.
• The behavior of the function: analyzing how \(z\) changes as \((x, y)\) changes.

Logarithmic Functions

Logarithmic functions are another important class of functions in calculus and beyond. A logarithm, expressed as \(\ln(x)\) for natural logs, asks the question: "to what power must the base, \(e\), be raised to produce \(x\)?"
Logarithmic functions have specific properties:

• They are defined only for positive values of \(x\), meaning the input to a logarithm must be greater than zero.
• They grow slower than any polynomial function, making them useful in modeling situations where growth decreases over time, such as in some population models or in the decay of radioactive materials.

In functions like \(g(x, y) = \ln(x^2 - y)\), it is crucial to ensure the expression within the logarithmic function is positive.
The logarithmic nature affects the domain of such functions significantly because it requires setting up an inequality that defines valid inputs.