Problem 7 - Calculus 3

Question

For the following exercises, find the domain of the function. $$ f(x, y)=4 \ln \left(y^{2}-x\right) $$

Step-by-Step Solution

Answer

The domain is \((x, y) \mid x < y^2\).

Step 1: Understand the Function

The function is given as \(f(x, y) = 4 \ln(y^2 - x)\). It involves a natural logarithm \(\ln\), which is defined only for positive arguments. Thus, we need \(y^2 - x > 0\) for \(f(x, y)\) to be defined.

Step 2: Solve the Inequality

We need to solve the inequality \(y^2 - x > 0\). Rearranging the terms gives \(x < y^2\). This inequality determines the set of \((x, y)\) pairs for which the function is defined.

Step 3: Determine the Domain

The domain of the function is the set of all \((x, y)\) pairs for which \(x < y^2\). Therefore, the domain in set notation is \(\{(x, y) \mid x < y^2\}\). This means for any real number \(y\), \(x\) must be less than \(y^2\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a mathematical function that gives us the power to which the base \(e\) must be raised to produce a given number. The constant \(e\) is an irrational number approximately equal to 2.71828. Natural logarithms are used widely in mathematics, especially in calculus, because of their convenient differentiation properties.

In our context, we have the function \(f(x, y) = 4 \ln(y^2 - x)\). For this function to be valid, the input to the natural logarithm (that is, \(y^2 - x\)) must be greater than zero:

• \(\ln\) is defined only for positive numbers.
• If the inside expression of \(\ln\) is negative or zero, \(f(x, y)\) is not defined.

This is a critical concept because it directly impacts the domain of our function. Understanding how \(\ln\) operates helps us determine the valid inputs for our function where it is mathematically defined.

Inequalities

Inequalities are mathematical expressions involving symbols like \(<\), \(>\), \(\leq\), and \(\geq\). They describe a range of possible values rather than a specific value.

In our case, we solved the inequality \(y^2 - x > 0\). This inequality shows the relationship between \(x\) and \(y\) that keeps the logarithmic function defined and valid.

Here's how you solve it:

• Start with the inequality \(y^2 - x > 0\).
• Rearrange the terms to find \(x < y^2\).

This tells us for every value of \(y\), \(x\) must be less than \(y^2\). This condition sets the domain of our function, determining where the function \(f(x, y)\) is ready to use. Solving inequalities is a fundamental skill in dealing with domains of logarithmic and other types of functions.

Function of Two Variables

Functions of two variables, like \(f(x, y)\), depend on two independent inputs, \(x\) and \(y\). The outputs are determined by these two variables together. Imagine a surface in 3D space that the function describes - height at any point \((x, y)\) is given by \(f(x, y)\).

For our given function \(f(x, y) = 4 \ln(y^2 - x)\), both \(x\) and \(y\) affect the outcome.

• The function is defined only for pairs \((x, y)\) satisfying the condition \(x < y^2\).
• This forms a region in the 2D \(xy\)-plane where the function remains valid and can be calculated.

By understanding the role of each variable, and how they interact within inequalities, we decipher the domain of a function with two variables. This concept is vital in multi-variable calculus as it leads to further studies in surface plotting and optimization problems in higher dimensions.