Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 1 - Section 1.8 - Inverse Functions - Exercise Set - Page 271: 67

Answer

See the explanation below.

Work Step by Step

(a) Firstly consider the Horizontal Line Test. The horizontal line drawn would intersect the graph at only one point. This shows that there is only one input for one output. Therefore, the graph has an inverse which is a function. (b) The x-axis shows the number of people and the y-axis shows probability of two people in a room sharing same birthday. Since, 0.25, 0.5, 0.7 are values of probability, we need to determine values of ${{f}^{-1}}\left( y \right)$. $f\left( x \right)=y$ So, take the inverse of the above function $f\left( x \right)=y$ ${{f}^{-1}}\left( f\left( x \right) \right)={{f}^{-1}}\left( y \right)$ Again ${{f}^{-1}}\left( f\left( x \right) \right)=x$ Therefore, $x={{f}^{-1}}\left( y \right)$ Thus, from the graph we obtain ${{f}^{-1}}\left( 0.25 \right)=15,\ {{f}^{-1}}\left( 0.5 \right)=22,\ {{f}^{-1}}\left( 0.7 \right)=30$ Practically, the function indicates that ${{f}^{-1}}\left( y \right)$ represents the number of people sharing a birthday corresponding to probability y. So, when probabilities will be an independent variable, the number of people sharing a birthday in a room will depend on these probabilities. Thus, the value of ${{f}^{-1}}\left( 0.25 \right),\ {{f}^{-1}}\left( 0.5 \right),\ {{f}^{-1}}\left( 0.7 \right)$ in practical terms is ${{f}^{-1}}\left( 0.25 \right)=15,{{f}^{-1}}\left( 0.5 \right)=22,{{f}^{-1}}\left( 0.7 \right)=30$.
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