Problem 17 - Calculus Early Transcendentals

Question

Find the domain of the following functions. $$g(x, y)=\ln \left(x^{2}-y\right).$$

Step-by-Step Solution

Answer

Question: Determine the domain of the function \(g(x, y) = \ln(x^2 - y)\). Answer: The domain of the function \(g(x, y) = \ln(x^2 - y)\) is the set \(\{(x, y) \in \mathbb{R}^2 \,|\, y < x^2\}\).

Step 1: Determine where the natural logarithm is defined

The natural logarithm function, \(\ln(u)\), is defined for positive values of \(u\). So, the function \(g(x, y) = \ln(x^2 - y)\) is defined for \((x, y)\) such that \(x^2 - y > 0\).

Step 2: Solve the inequality \(x^2 - y > 0\)

We need to find all values of \((x, y)\) for which \(x^2 - y > 0\). This inequality can be rewritten as: $$y < x^2$$

Step 3: Determine the domain of the function

The domain of the function is the set of all possible values of \((x, y)\) for which the function is defined. From step 2, we know that the function is defined for values of \((x, y)\) satisfying \(y < x^2\). All real values of x are allowed since \(x^2\) will always be non-negative. Therefore, the domain of the function \(g(x, y) = \ln(x^2 - y)\) is the set: $$\{(x, y) \in \mathbb{R}^2 \,|\, y < x^2\}$$

Key Concepts

Understanding Inequality

Understanding inequalities is essential when determining the domain of functions. When dealing with inequalities, you determine the range of values for which a particular expression holds true.
For example, in the given problem, we explore the inequality:

• \(x^2 - y > 0\)

This inequality states that the quantity \(x^2 - y\) must be greater than zero. To solve this, you can rearrange it to:

• \(y < x^2\)

This tells us that for each value of \(x\), \(y\) must be less than \(x^2\). Solving inequalities often leads to finding ranges or intervals of values that make the expression valid. Finding the solutions to inequalities is crucial when defining domains for functions like natural logarithms, ensuring that the functions operate within permissible values. These inequalities help to ensure that they work within their naturally defined scope.

The Natural Logarithm

The natural logarithm, represented as \(\ln(u)\), is a fundamental mathematical function. It is only defined for positive values of its argument, \(u\). This criterion is essential when considering the domain of logarithmic functions.
For the function \(g(x, y) = \ln(x^2 - y)\), it means that the expression inside the logarithm, \(x^2 - y\), must always be positive.
Why is this the case? The natural logarithm, inherently, cannot take zero or negative inputs. Attempting to calculate a logarithm of a non-positive value would result in undefined or complex values.
Therefore, ensuring that \(x^2 - y > 0\) guarantees the function operates in its defined range.

• This ensures the function output is meaningful and valid, relating directly to the inequality solved earlier.

Multivariable Functions

Multivariable functions are those that have more than one input variable, such as \(g(x, y)\). These functions map a set of two or more variables into a one-dimensional output. Understanding the domain of such functions requires considering restrictions on all variables involved.
For \(g(x, y) = \ln(x^2 - y)\), the domain involves understanding how both \(x\) and \(y\) influence the function's validity.

• Here, \(x\) can take any real number, but \(y\) must always be less than \(x^2\) for the logarithm to be defined.

This concept is crucial when working with functions of several variables because it provides insights into how the inputs combine and interact.
By understanding multivariable functions and their domains, one can visualize or determine for which combinations of \(x\) and \(y\) the function remains valid, reinforcing the need for both mathematical and spatial reasoning when dealing with complex functions.