Calculus: Early Transcendentals 8th Edition

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

Chapter 1 - Review - Concept Check: 12

Answer

(a) One to one function is the mapping such that $$f(x_1)=f(x_2)\Rightarrow x_1=x_2.$$ If the arbitrary horizontal line intersects the graph of the function once at most then the function is one-to-one. (b) $$f(x)=y\Rightarrow x=f^{-1}(y).$$ We obtain the graph of $f^{-1}$ by reflecting the graph of $f$ about the line $y=x$.

Work Step by Step

(a) One to one function is the mapping such that only one element of the domain $x$ maps to a certain element $y$ of the range i.e. $$f(x_1)=f(x_2)\Rightarrow x_1=x_2.$$ To tell from the graph ve can use the horizontal line test. If arbitrary horizontal line intersects the graph of the function once at most then the function is one-to-one since then only one value of $x$ corresponds to the certain value of $y$. (b) If the element $x$ of the domain maps by $f$ to $y$ then $y$ maps to $x$ by $f^{-1}$ i.e. $$f(x)=y\Rightarrow x=f^{-1}(y).$$ We obtain the graph of $f^{-1}$ by reflecting the graph of $f$ about the line $y=x$.
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