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Chapter 12: Problem 12

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

Short Answer

Expert verified
Answer: The domain of the function is any real values for x and y, or $${(x, y) \in \mathbb{R}^2}$$.

Step by step solution

01

Analyze the function

For our function $$f(x, y) = \cos{(x^2 - y^2)}$$, we have the cosine function applied to the expression $$(x^2 - y^2)$$. The domain of the cosine function is any real number, so we need to determine the domain of the expression inside the cosine function, $$(x^2 - y^2)$$.
02

Identify the domain of the expression

Since both x and y can take any real values and their squares, $$x^2$$ and $$y^2$$, are also real numbers, the expression $$(x^2 - y^2)$$ can also take any real value. As a result, there are no limitations on the values x and y can take.
03

Determine the domain of the function

Since the expression inside the cosine function can take any real value and the cosine function itself can accept any real number as an input, the domain of our function is any real values for x and y. In interval notation, we can represent the domain as $$(-\infty, \infty) \times (-\infty, \infty)$$ or in set notation, we can represent it as $${(x, y) \in \mathbb{R}^2}$$. To summarize, the domain of the function $$f(x, y) = \cos{(x^2 - y^2)}$$ is any real values for x and y, or $${(x, y) \in \mathbb{R}^2}$$.

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