Problem 13 - Calculus and its applications

Question

Determine the domain of each function of two variables. $$ g(x, y)=\frac{1}{y+x^{2}} $$

Step-by-Step Solution

Answer

The domain is \((x, y) \in \mathbb{R}^2 \) where \( y \neq -x^2 \).

Step 1: Understand the Function

The function presented is \( g(x, y) = \frac{1}{y + x^2} \). This is a rational function, meaning it is the ratio of two expressions. Our interest is in identifying any restrictions or limitations that affect the domain.

Step 2: Identify Potential Restrictions

Rational functions are undefined when their denominators are zero, as division by zero is undefined. Here, the denominator is \( y + x^2 \). We need to identify when this expression equals zero.

Step 3: Set Denominator to Zero

Set the denominator \( y + x^2 \) to zero: \[ y + x^2 = 0 \]. This equation shows us where the function \( g(x, y) \) is undefined.

Step 4: Solve the Equality

Rearrange the expression to find when it equals zero: \[ y = -x^2 \]. This tells us that for any point \((x, y)\) where \(y = -x^2\), the function \(g(x, y)\) is undefined.

Step 5: State the Domain

The domain of the function is all \((x, y)\) pairs except those where \(y + x^2 = 0\). Thus, the domain is the set of all \( (x, y) \) such that \( y eq -x^2 \). In interval notation, the domain can be expressed as: \[ \{ (x, y) \in \mathbb{R}^2 \mid y eq -x^2 \} \].

Key Concepts

Rational Functions

A rational function is a type of function that can be expressed as the ratio of two polynomials. Simply put, it's similar to a fraction, but instead of integers in the numerator and denominator, we're dealing with entire polynomial expressions. For example, in the function \( g(x, y) = \frac{1}{y + x^2} \), the numerator is the constant \(1\), and the denominator is the polynomial \(y + x^2\). The key feature of rational functions is their behavior when the denominator equals zero. This results in the function being undefined at those points. Rational functions can create interesting curve shapes and behaviors, especially around these undefined points where they can have asymptotes or holes. These characteristics make studying rational functions both challenging and intriguing.

Functions of Two Variables

Functions of two variables, like \( g(x, y) \), are an extension of the familiar single-variable functions. They take two inputs instead of one, mapping any

• Pair of real numbers \((x, y)\) to a value in the range.
• Function \( g(x, y) = \frac{1}{y + x^2} \) is an example.

These functions are fundamentally represented in a 3D space, with the x and y axes representing the two inputs, and the z-axis showing the output value. In a geometric context, these functions can be visualized as surfaces where every point \((x, y)\) corresponds to a value \(g(x, y)\) on the surface. This visualization can provide insights into behaviors such as peaks, valleys, and undulations across the surface.

Domain Restrictions

The domain of a function defines all the allowable input values. Identifying domain restrictions is crucial as it helps us understand where a function can run into problems, typically where it might become undefined. In rational functions, as seen in \( g(x, y) = \frac{1}{y + x^2} \), domain restrictions often occur where the denominator might be zero. Because division by zero is undefined, these points cannot be included in the domain.

• For \( g(x, y) \), the expression \( y + x^2 \) must not equal zero.
• By setting \( y + x^2 = 0 \) and solving for \( y \), we find this occurs when \( y = -x^2 \).

This means the domain includes all pairs except for when \( y + x^2 = 0 \). Therefore, we exclude the line described by \( y = -x^2 \) from our set of permissible inputs. Recognizing domain restrictions enables better understanding and analysis of how functions behave across different inputs.