Problem 24 - Calculus an Applied Approach

Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ g(x, y)=\frac{1}{x-y} $$

Step-by-Step Solution

Answer

The region \(R\) is the entire \(x-y\) plane except the line where \(x = y\).

Step 1: Identify Unallowed Values

Find the values where the function \(g(x, y)\) is undefined. For a fraction to be defined, the denominator must not be equal to zero. Here the denominator is \(x - y\), and it would be equal to zero where \(x = y\). So, the function is undefined at \(x = y\).

Step 2: Describe the region

Since the only values where the function is undefined are when \(x = y\), the function \(g(x, y)\) is defined for all other values. Hence, the region \(R\) in the \(x-y\) plane includes all points except those on the line \(x = y\). This is a diagonal line going through the origin and is not part of the domain of the function.

Key Concepts

Domain of a Function

In calculus, the concept of the domain of a function is vital. It represents all the possible inputs for which the function is defined or "works."
For a function involving two variables in the form of a fraction, like \( g(x, y) = \frac{1}{x - y} \), the domain is the set of all \(x\) and \(y\) values that do not cause any mathematical inconsistencies.
Most often, this means looking out for any division by zero, since division by zero is undefined in mathematics.

• The function is defined where its expression is valid.
• The domain excludes any points where the function is undefined.
To determine the domain for \( g(x, y) \), we need to ensure that the denominator \( x - y \) is not zero.
Therefore, the domain includes every point in the \(xy\)-plane except where \( x = y \).
This understanding is crucial for effectively evaluating the function across various limits.

Undefined Values

Understanding undefined values is essential when studying functions, especially those with fractional components. When a function is undefined, it means there's an input that does not produce a logical or real output.
For the function \( g(x, y) = \frac{1}{x - y} \), focus on the denominator of the fraction, \( x - y \). If we allow \( x = y \), the denominator becomes zero, making the function undefined due to division by zero.
This helps in pinpointing specific inputs that the function cannot accept.

• Check the denominator to avoid undefined regions.
• Identify these inputs to better outline the domain.

By identifying where \(x = y\), we see that these values form a line that \( g(x, y) \) cannot include as pictured in the solution, which affects its domain.

XY-Plane

The \(xy\)-plane is a two-dimensional graph used to visibly depict the domain of functions with two variables.
This plane is composed of all possible coordinate points \((x, y)\).
When assessing the domain of a function like \( g(x, y) = \frac{1}{x - y} \), the \(xy\)-plane helps visualize which points are included and which are excluded.
In this case, the function is undefined on the line \(x = y\), forming a diagonal line through the origin. Every other point in the \(xy\)-plane is part of the domain.

• Visualize the domain effectively.
• Understand how exclusion affects graph representation.

Using the \(xy\)-plane, we can easily see how the line \(x = y\) divides the plane, showing the domain of \( g(x, y) \) as every point not on this line, helping to avoid undefined values while understanding the function's behavior.