Problem 38 - Calculus 3

Question

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

Step-by-Step Solution

Answer

The domain is all real numbers: \((x, y) \in \mathbb{R}^2\).

Step 1: Identify the Structure of the Function

The given function is a composition of a square root and cosine function, specifically: \( f(x, y) = \cos \sqrt{x^2 + y^2} \). Understanding how these components work is crucial to finding the domain.

Step 2: Determine the Domain of the Square Root Function

The expression inside the square root, \( x^2 + y^2 \), must be non-negative (\( \geq 0 \)), as the square root function is undefined for negative values. Since \( x^2 + y^2 \geq 0 \) for all real \( x \) and \( y \), this part of the function is always defined.

Step 3: Consider the Cosine Function

The cosine function, \( \cos(\theta) \), is defined for all real numbers. Therefore, regardless of whether \( \sqrt{x^2 + y^2} \) is any non-negative real number, \( \cos \sqrt{x^2 + y^2} \) will be defined as well.

Step 4: Integrate Both Conditions for the Composite Function

Since both the square root and cosine functions are defined for all real \( x \) and \( y \) that satisfy \( x^2 + y^2 \geq 0 \), the composite function \( f(x, y) = \cos \sqrt{x^2 + y^2} \) is defined for all \( x \) and \( y \) in \( \mathbb{R}^2 \).

Key Concepts

Domain of a Function

When dealing with multivariable calculus, understanding the domain of a function is crucial. The **domain** refers to the set of all possible input values that a function can accept without causing any mathematical errors. Specifically, a function like \( f(x, y) = \cos \sqrt{x^2 + y^2} \) requires us to determine which values of \( x \) and \( y \) make the function valid.

• To find the domain, observe each component of the composite function.
• Ensure that operations like square roots do not involve negative numbers.
• Check that trigonometric functions are defined for the values in question.

By understanding the constraints of each function component, we can establish where the overall function is defined. For the explored function, the domain results in all real \( x \) and \( y \), meaning any point in the 2D plane \( \mathbb{R}^2 \).

Composite Functions

In the realm of multivariable calculus, a **composite function** is a function made up of multiple simpler functions. It's like stacking one function on top of another. For instance, in \( f(x, y) = \cos \sqrt{x^2 + y^2} \), the square root function is nested inside the cosine function.

• This setup is notated as \( f(g(h(x, y))) \), where each function affects the outcome of the next.
• Each function has its own domain, and assessing them step-by-step helps find the overall domain.

Breaking down composite functions into their inner and outer parts reveals how they interact. Recognizing the sequence helps us trace through the function, ensuring each step remains valid under defined mathematical rules. This approach assists in managing complex systems in calculus, establishing both the limits and the behaviors of the function.

Square Root Function

The **square root function** is fundamental in defining certain multivariable calculus problems. It is typically expressed as \( \sqrt{x^2 + y^2} \) in functions like \( f(x, y) = \cos \sqrt{x^2 + y^2} \).

• Critically, the square root must have a non-negative argument.
• This means \( x^2 + y^2 \geq 0 \). Quadratic expressions like \( x^2 + y^2 \) are naturally non-negative for all real \( x \) and \( y \).

Since negative numbers have no real square roots, a critical step in determining the domain is ensuring that the argument is never below zero. In this example, because \( x^2 + y^2 \) cannot be negative, the square root component is always defined. Hence, the presence of \( x^2 + y^2 \) ensures broad applicability across \( \mathbb{R}^2 \), enhancing the domain of the entire function.

Cosine Function

The **cosine function**, denoted as \( \cos(\theta) \), plays an important role in trigonometric calculus problems. It's a smooth, oscillating function defined for all real numbers, with outputs ranging between \(-1\) and \(1\).

• Cosine is unique as it faces no restrictions regarding its domain; \( \theta \) can be any real number.
• When paired with other functions, like a square root, check that the resulting input remains valid.

For the composite function \( f(x, y) = \cos \sqrt{x^2 + y^2} \), this means \( \cos \) will accept any non-negative argument provided by \( \sqrt{x^2 + y^2} \). The universality of the cosine function's domain simplifies many complex functions, as it effortlessly accommodates any real input brought forth by other function components.