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: 66


a) $g=\left\{ \left( \text{Zambia,}-\text{7}\text{.3} \right),\ \left( \text{Columbia,}-\text{4}\text{.5} \right),\ \left( \text{Poland,}-\text{2}\text{.8} \right),\left( \text{Italy,}-\text{2}\text{.8} \right),\ \left( \text{United States,}-\text{1}\text{.9} \right) \right\}$ b) ${{g}^{-1}}=\left\{ \left( -\text{7}\text{.3},\text{Zambia} \right),\ \left( -\text{4}\text{.5,Columbia} \right),\ \left( -\text{2}\text{.8,Poland} \right),\left( -\text{2}\text{.8,Italy} \right),\ \left( -\text{1}\text{.9,United States} \right) \right\}$ It is not a one-to-one function.

Work Step by Step

(a) In the above graph, the 5 countries are: Zambia, Columbia, Poland, Italy, and United States. According to each of these countries, the set of the negative numbers representing the average number of years that men in each country prefer women who are younger than themselves is as follows: $-7.3,\ -4.5,\ -2.8,\ -2.8,\ -1.9$ And the ordered pairs formed out of this data will be: $\left( \text{country,}\ \text{number of years in negative} \right)$=$\left( \text{Zambia, }-\text{7}\text{.3} \right)$$\left( \text{Columbia, }-\text{4}\text{.5} \right)$ $\left( \text{Poland, }-\text{2}\text{.8} \right)$$\left( \text{Italy, }-\text{2}\text{.8} \right)$$\left( \text{United States, }-\text{1}\text{.9} \right)$ $g=\left\{ \left( \text{Zambia,}-\text{7}\text{.3} \right),\ \left( \text{Columbia,}-\text{4}\text{.5} \right),\ \left( \text{Poland,}-\text{2}\text{.8} \right),\left( \text{Italy,}-\text{2}\text{.8} \right),\ \left( \text{United States,}-\text{1}\text{.9} \right) \right\}$ (b) The inverse function is obtained by the exchange of the domain and range of g. Thus, the inverse function is obtained as $\left( \text{number of years in negative, country} \right)$ Therefore, ${{g}^{-1}}=\left\{ \left( -\text{7}\text{.3},\text{Zambia} \right),\ \left( -\text{4}\text{.5,Columbia} \right),\ \left( -\text{2}\text{.8,Poland} \right),\left( -\text{2}\text{.8,Italy} \right),\ \left( -\text{1}\text{.9,United States} \right) \right\}$ For a function to be one-to-one, each value of the domain must have a unique value in the range. The value −2.8 in the domain corresponds to Poland as well as Italy in the range, so it is not a one-to-one function. Hence, the relation that is the inverse of g is as follows: ${{g}^{-1}}=\left\{ \left( -\text{7}\text{.3},\text{Zambia} \right),\ \left( -\text{4}\text{.5,Columbia} \right),\ \left( -\text{2}\text{.8,Poland} \right),\left( -\text{2}\text{.8,Italy} \right),\ \left( -\text{1}\text{.9,United States} \right) \right\}$ Hence, it is not a one-to-one function.
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