Word Count : 457

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### Function: The given function is \( f(x, y) = \frac{2xy}{x^2 + y^2} \). ### Domain: The domain of the function refers to all the pairs \( (x, y) \) for which the function is defined. To find the domain, we need to ensure that the denominator \( x^2 + y^2 \neq 0 \). The denominator is zero when \( x = 0 \) and \( y = 0 \), which corresponds to the origin. Therefore, the function is undefined at \( (0, 0) \). Domain: \[ \boxed{ \{ (x, y) \in \mathbb{R}^2 \mid (x, y) \neq (0, 0) \} } \] This means the domain is the entire plane except for the origin. ___ _________ ________ ______ ________ __________ ____.
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