Suppose you need to choose between two car rental companies. you plan on driving for 6 hours at a constant speed of 100km/h.

Question

Company 1 charges $1.50/km and a daily fee of $45.
Company 2 charges $1.65/km and a daily fee of $40.

What is the difference in price between the two companies? Use composite functions to determine your
answer.

Answer 1

 

Let k = # of kilometers driven

 

COMPANY 1:

 

    fixed fee = $45

    km charge = 1.50k   ($1.50 for each km driven)

    

    Cost function:  f(k) = 1.50k + 45

 

    Determine the distance driven:

 

    Rate • Time = Distance

    100 km/hr • 6 hr = 600 km

 

    Plug 600 into the Cost function

    f(600) = 1.50(600) + 45

                = $945

 

COMPANY 2:

    fixed fee = $40
    km charge = 1.65k ($1.65 for each km driven)

    Cost function: g(k) = 1.65k + 40

    Number of miles driven is 600 km (sane as for company 1)

    Plug 600 into the Cost function
    g(600) = 1.65(600) + 40
               = $1030

 

clearly, company 1 is the better deal (cheaper cost) ... f(600) < g(600)

 

The problem asked to use composite functions.

 

I will use

 

     f(k)  = 1.50k + 45     (company 1)

     g(k) = 1.65k + 40     (company 2)

 

(f - g)(k) = f(k) - g(k)

               = (1.50k + 45) - (1.65k + 40)

               = 1.50k + 45 = 1.65 - 40

               = -.15k + 5

(f - g)(600) = -.15(600) + 5 = -85

 

(g - f)(k) = g(k) - f(k)

              = (1.65k + 40) - (1.50k + 45)

              = 1.65k + 40 - 1.50k - 45

              = .15k - 5

(g - f)(600) = .15(600) - 5

                  = 85

 

The difference in cost is $85

 

 

Answer 2

Rafal,

 

The general form for composite functions (assume 2) looks like this:

 

(f ο g)x where f(x) and g(x) are the two functions.  A clearer way to write this is:

 

f(g(x)) where evaluating g(x) comes first and then using the value we get for g(x) in f(x).

 

So, for this problem, we need to have two functions that describe the situation.  First, we need to understand that to answer this question, we'll need to know the total distance traveled (function 1).  

 

g(x) = hours * (100km/h) = 6 * (100km/h) = 600km (same for both companies)

 

Now that we have evaluated g(x), we need to find f(x) (function 2).

 

f(x) = km * (cost/km) + daily fee

 

Then, for rental company 1:

 

f(g(x)) = 600 * ($1.50/km) + $45 = $900 + $45 = $945 (what rental company 1 charges)

 

For company 2:

 

f(g(x)) = 600 * ($1.65/km) + $40 =  $990 + 40 = $1030 (what rental company 2 charges)

 

I'd pick company 1

 

Please note that although we used x to indicate the variable in both functions, it doesn't represent the same thing in both; in g(x) it's hours; in f(x) it's km.

 

Hope this helps!