Problem 10 - Calculus 3

Question

For the following exercises, find the domain of the function. $$ f(x, y)=\frac{y+2}{x^{2}} $$

Step-by-Step Solution

Answer

The domain is all \((x, y)\) such that \( x \neq 0 \).

Step 1: Identify the Conditions for the Fraction to be Defined

For the function \( f(x, y) = \frac{y+2}{x^2} \) to be defined, the denominator must not be zero. Therefore, we need to ensure that \( x^2 eq 0 \).

Step 2: Solve the Denominator Inequality

The inequality \( x^2 eq 0 \) implies that \( x eq 0 \). This means that \( x \) can take any real value except zero.

Step 3: Determine the Final Domain

As long as \( x eq 0 \), there are no restrictions on \( y \). Therefore, the domain of the function is all pairs \((x, y)\) such that \( x eq 0 \).

Key Concepts

Exploring the Domain of a Function

Understanding the domain of a function is like figuring out the complete set of values for which the function is properly defined. For multivariable functions, it's important to consider all variables involved. In our exercise, the function is given by:\[ f(x, y) = \frac{y+2}{x^2} \]To find where this function is valid, we need to determine the values of \((x, y)\) for which the function does not run into any mathematical errors, such as division by zero. The domain in this scenario encompasses all possible values of \((x, y)\) for which the denominator is not zero. Let’s delve deeper into this aspect in the next section.

Understanding Denominator Inequality

When dealing with fractions in mathematics, ensuring that the denominator is never zero is crucial. This ensures the function does not yield undefined values. In our given function:\[ f(x, y) = \frac{y+2}{x^2} \]The denominator is \(x^2\). For this function to remain defined:- It is essential that \(x^2 eq 0\).- This inequality simplifies down to \(x eq 0\).Thus, the inequality tells us \(x\) can take any nonzero value on the number line. This constraint is solely related to the variable \(x\), leaving \(y\) free from any restrictions imposed by the denominator. Moving forward, we'll see how this contributes to identifying the domain.

Discovering Real Values in Domains

Real values are those that include every number on the number line, encompassing positive, negative, and decimal numbers. In the context of the domain for \(f(x, y)\), \(x\) can be any real number except zero due to the restriction created by the denominator inequality. On the other hand, \(y\) is unrestricted.Let's define the domain based on real values:- \(x\) can be any real number (\(\mathbb{R}\)) except zero.- \(y\) can be any real number (\(\mathbb{R}\)) without restriction.This implies that the domain is represented by all ordered pairs \((x, y)\) such that \(x eq 0\). This way, the function operates without dividing by zero and taps into the full spectrum of real numbers. This foundational understanding of real values is instrumental in analyzing functions effectively.