Question

2.)The resale value of a textbook decreases by 25% with each previous owner. A new textbook is sold for $85. Which function represents the resale value of the textbook after x owners?
f(x) = 85(1 – 0.25)^x

f(x) = 85(1 + 0.25)^x

f(x) = 85(0.25)^x

f(x) = (85 – 0.25)^x

Answer 1

Since they value of the book is decaying, we are going to adapt the standard exponential decay function to solve this.

Exponential decay function: $f(x)=a(1-b)^x$

where

$f(x)$ is the final amount remaining after $x$ years of decay

$a$ is the initial amount

$b$ is the decay rate in decimal form

$x$ is the time in years

Now, let's adapt the equation to suit our needs

Since the value of the book is decaying with the number of owners and not with time, we are going to let $x$ be the number of previous owners instead of time. So now $f(x)$ will represent the final cost of the book after $x$ owners, and $a$ will be the initial cost of the book.

Exponential decay function: $f(x)=a(1-b)^x$

where

$f(x)$ is the final cost of the book after $x$ owners

$a$ is the initial cost of the book

$b$ is the decay rate in decimal form

$x$ is the number of previous owners

Now, we know from our problem that the initial cost of the book is $85, so $a=85$. We also know that the resale value of a textbook decreases by 25% with each previous owner; to convert the decay rate (25%) to decimal form, we are going to divide it by 100%:

$b=\frac{25}{100}$

$b=0.25$

We have everything we need so let's replace the values in our exponential decay equation:

$f(x)=a(1-b)^x$

$f(x)=85(1-0.25)^x$

We can conclude that the correct answer is A) f(x) = 85(1 – 0.25)^x

Answer 2

Given that the original value is $85, therefore at x = 1, f(x) = $85. It is given that the price decreases by 25% with each previous owner. This means that at the 2nd owner, the value is 75% of $85, which is $63.75. Finding an equation (from the choices) which fits these parameters, the answer is f(x) =85(1-.025)^x (or A).