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 228: 101

Answer

$(a)$ $f(x)=0.85x$ $(b)$ $g(x)=x-1000$ $(c)$ $H=0.85x-850$ $(d)$ $H^{-1}=\frac{20}{17}x+1000$ $(e)$ $\approx\$16284.12$ It's the original price of the car for given purchase price $\$13000$

Work Step by Step

$(a)$ If $x$ is the price of a car, then the amount actually paid for the car will be $85\%$ of actual price. That is represented by the following function: $f(x)=0.85x$ $(b)$ If only $\$1000$ rebate is applied, then we simply subtract the rebate from the price of the car: $g(x)=x-1000$ $(c)$ $f◦g=f(g(x)) = 0.85(x-1000)=0.85x-850$ $(d)$ For $H^{-1}$, we first write it in terms of $x$ and $y$ and then replace $x$ by $y$ and vice versa. Then find $y$ in terms of $x$: $y=0.85x-850$ $x=0.85y-850$ $0.85y=x+850$ $y=\frac{x+850}{0.85}$ $y=\frac{100x+85000}{85}$ $y=\frac{100x}{85}+\frac{85000}{85}$ $y=\frac{20}{17}x+1000$ $H^{-1}=\frac{20}{17}x+1000$ $(e)$ $H^{-1}(13000)=\frac{20}{17}\times13000+1000\approx16284.12$ It's the original price of the car for given purchase price $\$13000$
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