Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 2 - Review - Concept Check - Page 231: 15

Answer

$(a)$ One-to-one function is a function whose each element in its range corresponds to the only one element in its domain. $(b)$ Graphically, we can use the Horizontal Line Test to determine whether a function is one-to-one or not. $(c)$ The range and the domain will switch each other. $f^{-1}$ will have domain $B$ and range $A$. $(d)$ To find inverse of a function, we have to follow the next steps: $1.$ Write the function in terms of $y$ and $x$ ($f(x)=y$) $2.$ Switch $y$ by $x$ and vice versa $3.$ simplify the equation for $y$ $f^{-1}=\frac{x}{2}$ $(e)$ If we are given the graph of $f$ we simply reflect it about $y=x$ line and we will get $f^{-1}$

Work Step by Step

$(a)$ One-to-one function is a function whose each element in its range corresponds to the only one element in its domain. $(b)$ Graphically, we can use the Horizontal Line Test to determine whether a function is one-to-one or not. The Horizontal Line Test means to visually imagine infinite amount of horizontal lines. There has to be no horizontal line that crosses the graph of a function more than one time. $(c)$ The range and the domain will switch each other. $f^{-1}$ will have domain $B$ and range $A$. $(d)$ To find inverse of a function, we have to follow the next steps: $1.$ Write the function in terms of $y$ and $x$ ($f(x)=y$) $2.$ Switch $y$ by $x$ and vice versa $3.$ simplify the equation for $y$ $f(x)=2x$ $y=2x$ $x=2y$ //Switch $y=\frac{x}{2}$ $f^{-1}=\frac{x}{2}$ $(e)$ Inverse of a function is reflection of the function about $y=x$ line. So, if we are given the graph of $f$ we simply reflect it about $y=x$ line and we will get $f^{-1}$
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