Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 7 - Functions - Exercise Set 7.2 - Page 414: 17

Answer

f is one-to-one

Work Step by Step

$f(x)=\frac{3x-1}{x}\\ A\,function\,\,f: \mathbb{R}^\ast \rightarrow \mathbb{R}^\ast \,is\,\,one-to-one\,\Leftrightarrow \\ \forall \,\,x_{1}\,and\,x_{2}\,\,in\,\,\mathbb{R}^\ast \,\,if\, f(x_{1}) = f(x_{2})\,\,then\,x_{1} = x_{2}.\\ let\,f(x_{1})=f(x_{2})\\ \Rightarrow \frac{3x_{1}-1}{x_{1}}=\frac{3x_{2}-1}{x_{2}}\\ \Rightarrow (3x_{1}-1)x_{2}=(3x_{2}-1)x_{1}\\ \Rightarrow 3x_{2}x_{1}-x_{2}=3x_{2}x_{1}-x_{1}\\ \Rightarrow -x_{1}=-x_{2}\\ \Rightarrow x_{1}=x_{2}\\ therefore\,f\,is\,one-to-one $
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