Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Section 1.4 - Exponential Functions - 1.4 Exercises - Page 53: 19


a) $(-\infty, -1)\cup(-1, 1)\cup(1,\infty)$ b) $(-\infty, \infty)$

Work Step by Step

To find the domain, we look for which values of $x$ the function is undefined. For both a) and b), it is when the denominator equals $0$. a) $1-e^{1-x^2}=0$ $1=e^{1-x^2}$ $\ln(1)=\ln(e^{1-x^2})$ $0=\ln(e^{1-x^2})$ Using a log property (where $\ln{x^y}=y\ln{x}$): $0=(1-x^2)\ln(e)$ Remembering that $\ln(e)=1$ $1=x^2$ $x=-1,\thinspace 1$ So for a), the domain is $(-\infty, -1)\cup(-1, 1)\cup(1,\infty)$ b) $e^{cos{x}}=0$ There is no $x$ value where this equation holds true, so the domain is all real numbers.
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