Chapter 1
Relationships





1.5 The Algebra of Functions

1.5.2 Inverse Functions: Car Payments

On the preceding page we stressed the importance of function-as-object because we will often do operations to functions. Here we single out one operation for special attention, namely the operation of inverting or undoing a function. As with the other concepts in this chapter, this will be a frequently recurring theme throughout the course. We begin with an example that you may find of personal interest, namely, car payments.

The monthly payment on a new car depends on four factors:

  1. the price of the car,
  2. the size of the down payment,
  3. the annual interest rate, and
  4. the term of the loan.

These quantities are related (for reasons we will see much later) by the formula

p = ( P - D )   r ⁢   ( 1 + r ) n ( 1 + r ) n - 1 ⁢ ,

where

`p` is the monthly payment,
`P` is the price of the car,
`D` is the down payment,
`r` is the monthly interest rate (as a decimal fraction), and
`n` is the number of months required to pay back the loan.

Note 1Note 1 – How much later?

The interest rate and the duration of the loan are usually determined by the seller, and the down payment is agreed on by the buyer and seller. Once these three quantities are set, the monthly payment and the price of the car are the only true variables in the formula. The price of the car determines the monthly payment, so price is usually thought of as the independent variable and the monthly payment is a function of the price.

Activity 2

  1. Suppose you want to buy a `\$`19,000 car, and you have `\$`2700 for a down payment. The annual interest rate, because the dealer is trying to move cars off the lot, is a very favorable 3.9% (equivalent to 0.325% per month), and the term of the loan will be 48 months. What does the dealer tell you your monthly payment will be? (You may use our pop-up calculator or one of the computer algebra files for assistance with this calculation and the one in part b.)
  2. Your income and your other expenses limit you to a monthly payment of `\$`280. Can you afford the `\$`19,000 car? How expensive a car can you afford?

Comment 2 Comment on Activity 2

Activity 3

  1. Don't substitute `p=280` in the formula, or the other known values. Solve for `P` to find a formula for the inverse function, i.e., for `P` as a function of `p` (with `D`, `r`, and `n` still assumed to be constants).
  2. When you substitute the known values for `D`, `r`, and `n` in the original formula, the resulting formula expresses `p` as a linear function of `P`. What is the slope of that linear function?
  3. In part a, you found that `P` is also a linear function of `p`. When you substitute the known values for `D`, `r`, and `n` in part b, what slope do you get?
  4. What is the relationship between the two slopes in parts b and c?

Comment 3 Comment on Activity 3

In part (d) of Activity 3, you should have found that the slopes of the inverse linear functions are reciprocals of each other. This is no accident. In fact, for any monthly interest rate `r`, the slope of the payment function is

`(rtext[(]1+rtext[)]^n)/(text[(]1+r text[)]^n-1)`,

and the slope of the price function is

`(text[(]1+r text[)]^n-1)/(rtext[(]1+r text[)]^n)`,

clearly reciprocals of each other.

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