Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 13 - Functions of Several Variables - 13.1 Exercises - Page 876: 20


The domain is given by $$(x,y)\in (-\infty,\infty)\times (-\infty,\infty) =\mathbb{R^2}.$$ The range is the interval $(0,\infty)$.

Work Step by Step

We have that $$f(x,y) = e^{xy}$$ The exponential is defined for every real number so the product $xy$ can be any real number which means that both $x$ and $y$ can take any real value i.e. $$x\in(-\infty,\infty)=\mathbb{R},\quad y\in(-\infty,\infty)=\mathbb{R};$$ $$(x,y)\in (-\infty,\infty)\times (-\infty,\infty) =\mathbb{R^2}.$$ As for the range, the values of the exponent with the positive basis (as $e$ is) are always positive. In this case, since $xy$ will take all real values, then the value of $e^{x,y}$ will cover all positive reals i.e. the range is $(0,\infty)$.
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