Problem 12 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The domain of the function \(f(x, y)\) is all real numbers for both x and y, represented as \(\{(x, y) \in \mathbb{R}^2\}\).

Step 1: Identify the function

The given function is $$f(x, y) = \cos\left(x^2 - y^2\right).$$

Step 2: Analyze the cosine function

The cosine function is defined for all real numbers, meaning that it accepts any real number as its argument.

Step 3: Analyze the argument \(x^2 - y^2\)

Both \(x^2\) and \(y^2\) are non-negative for all real numbers x and y. Their difference, \(x^2 - y^2\), can be any real number as both x and y can take any real number.

Step 4: Determine the domain of the function

Since the cosine function is defined for all real numbers and the argument \(x^2 - y^2\) can be any real number, the domain of the function is unrestricted. The function is defined for all real numbers x and y.

Step 5: Write the domain in proper notation

The domain of the function \(f(x, y)\) is all real numbers for both x and y. We can write this domain as $$\text{Domain}(f(x, y)) = \{(x, y) \in \mathbb{R}^2\}.$$

Key Concepts

Domain of a Function

The domain of a function is essential in calculus. It refers to the set of all possible inputs for which the function is defined.

• For two-variable functions like \( f(x, y) = ext{some expression involving } x ext{ and } y \), we examine if substitute values will lead to a defined result.
• In general, a function can be undefined if substituting values cause division by zero or involve taking roots of negative numbers in the real number system.
• However, functions that involve operations like addition, multiplication, and trigonometric functions, typically have broader domains.

By understanding the domain, you can determine how far your analysis can extend across the input values. This is crucial in applications, modeling real situations, or solving mathematical problems.
For instance, with the function \( f(x, y) = \cos(x^2 - y^2) \), the domain encompasses all possible real-value pairs \((x, y)\) without restrictions.

Cosine Function

The cosine function, represented as \( \cos(\theta) \), is a fundamental trigonometric function with certain characteristics:

• Values range from -1 to 1, inclusive, for the cosine output.
• It is periodic with a period of \( 2\pi \), meaning it repeats its pattern after every \( 2\pi \) units along the x-axis.
• The cosine function is defined for all real numbers, so any real number can be an input to give a corresponding output between -1 and 1.

For the function \( f(x, y) = \cos(x^2 -y^2) \), because the cosine function itself handles any real number argument smoothly, the focus shifts to understanding its effect on the particular expression \( x^2 - y^2 \) provided as its input.

Real Numbers

Real numbers are a key concept, forming a complete set of numbers without any gaps on the number line. They include:

• Positive and negative whole numbers (integers).
• Fractions and decimals (rational numbers).
• Numbers that cannot be expressed as fractions, like \( \sqrt{2} \) or \( \pi \) (irrational numbers).

Real numbers are used in multivariable calculus to explore functions of several variables because of their broad scope and applicability in various mathematical contexts.
In our function \( f(x, y) = \cos(x^2 - y^2) \, \) both \( x \) and \( y \) are based in the set of all real numbers. Their operations, such as the squaring \((x^2 \) and \( y^2)\), and the difference between them \( x^2 - y^2 \), all remain within real numbers.