Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Section 7.1 - Inverse Functions and Their Derivatives - Exercises 7.1 - Page 373: 47

Answer

$ a.\quad f^{-1}(x)=x-1$. See image for graphs. $ b.\quad$ $f^{-1}(x)=x-b$ The graph of $f^{-1}$ is parallel to the graph of $f$, with the y-intercept of $-b.$ $ c.\quad$ They will be reflections of the original line, with regard to the line $y=x$

Work Step by Step

$ a.\quad$ Find the inverse: Set $y=f(x)$. Interchange $x$ and $y$: $y=x+1$ $x=y+1$ ... solve for y $y=x-1$ ... replace y with $f^{-1}(x)$ $f^{-1}(x)=x-1$ See image for graphs. $ b.\quad$ Finding the inverse, set y=f(x). Interchange and y $y=x+b$ $x=y+b$ ... solve for y $y=x-b$ ... replace y with $f^{-1}(x)$ $f^{-1}(x)=x-b$ The graph of $f^{-1}$ is parallel to the graph of $f$, with the y-intercept of $-b.$ $ c.\quad$ According to the result of (b), they will be parallel and have the opposite y-intercept. In other words, they will be the mirror image of the original line, with regard to the line $y=x.$ (Reflect the point (a,b) over the line $y=x$ by plotting (b,a).)
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