Problem 7 - Calculus Volume 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 \( \{(x, y) \mid y^2 > x\} \).

Step 1: Understand the Function

The function given is \( f(x, y) = 4 \ln(y^2 - x) \). The function involves a natural logarithm, \( \ln \), which is only defined for positive arguments.

Step 2: Set the Argument Greater than Zero

To find where \( \ln(y^2 - x) \) is defined, set its argument to be greater than zero: \( y^2 - x > 0 \).

Step 3: Solve the Inequality

Rearrange the inequality from Step 2: \( y^2 > x \). This inequality must hold true for \( f(x, y) \) to be defined.

Step 4: Determine the Domain

The domain of the function \( f(x, y) \) consists of all pairs \( (x, y) \) for which \( y^2 > x \). This can be represented as the set of all \( (x, y) \) such that for each \( y \), \( x < y^2 \).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental mathematical function extensively used in calculus and analysis. It is defined only for positive real numbers. This means that for \( \ln(x) \) to return a real number, \( x \) must be greater than zero. This restriction plays a crucial role in determining the domain of functions involving the natural logarithm.

Here are some important properties of the natural logarithm:

• \( \ln(1) = 0 \) because the exponential function \( e^0 = 1 \).
• \( \ln\) is undefined for \( x \leq 0 \).
• As \( x \) approaches zero from the positive side, \( \ln(x) \) tends to negative infinity.

When working with functions that involve \( \ln \), it is crucial to first ensure its argument is positive. In the context of the original exercise, \( \ln(y^2 - x) \) mandates that \( y^2 - x > 0 \), thus shaping the domain of the function.

Solving Inequalities

Inequality solving is a fundamental concept in algebra, crucial for determining the domain of functions where certain restrictions apply. In the case of the original exercise, we deal with the inequality \( y^2 - x > 0 \). Solving this inequality means finding all the values \( x \) and \( y \) can take so that the inequality holds true.

Here are some steps to solve inequalities effectively:

• Isolate the variable expression on one side of the inequality.
• Manipulate the inequality just like an equation, remembering that multiplying or dividing by a negative number will reverse the inequality sign.
• Identify the solution set, often represented graphically on a number line or through an interval.

In this example, the solution \( y^2 > x \) means any \( (x, y) \) pair such that each \( x \) is less than \( y^2 \) is part of the domain. This inequality ensures the function remains valid across its domain.

Exploring Multivariable Functions

Multivariable functions take two or more variables as inputs, providing a more complicated yet richer canvas for expressing relationships. The function \( f(x, y) = 4 \ln(y^2 - x) \) is one such function, dependent on both \( x \) and \( y \).

Characteristics of multivariable functions:

• They can represent surfaces in three-dimensional space when plotted, where each point on the surface corresponds to an input \( (x, y) \).
• The domain of a multivariable function is determined by ensuring each point meets any set conditions or restrictions.
• These functions are intrinsic to fields such as physics, engineering, and economics, where multiple influencing factors are considered.

In our function, the domain is defined by the condition \( y^2 > x \). This means that for any fixed value of \( y \), \( x \) must be less than the square of \( y \). Understanding multivariable functions requires considering how each variable influences the output and ensuring any given pair \( (x, y) \) meets the domain's conditions to yield a meaningful result.