Precalculus: Mathematics for Calculus, 7th Edition

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

Chapter 2 - Section 2.8 - One-to-One Functions and Their Inverses - 2.8 Exercises - Page 227: 92

Answer

(a) See the image below. (b) Yes, the graph clearly shows that $f$ and $f^{-1}$ are the same function. (c) $f^{-1}(x)=\frac{x+3}{x-1}$

Work Step by Step

(a) See the image above. (b) Due to the definition of the inverse function, we know that the graph of an inverse function is a reflection of a graph about $y=x$ axis. As we can clearly see from the image above, the graph is symmetrically divided by $y=x$ axis. So, yes $f$ and $f^{-1}$ are the same function. (c) To calculate the inverse function, we will do the following: $f(x)=\frac{x+3}{x-1}$ First we have to write it down in terms of $y$ and $x$: $y=\frac{x+3}{x-1}$ Then replace $y$ by $x$ and vice versa: $x=\frac{y+3}{y-1}$ And at last solve it for $y$ $x(y-1)=y+3$ $xy-x=y+3$ $xy-y=x+3$ $y(x-1)=x+3$ $y=\frac{x+3}{x-1}$ $f^{-1}(x)=\frac{x+3}{x-1}$
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