Pre Calc Word Problem.. HELP!

Question

 A certain calling card gives the following rate to call Malaysia. There is a connection fee of $0.39 which allows you to connect for up to (but not including) one minute. For each additional minute or part of a minute, there is a $0.24 cent charge. (For example, 1 minute up to but not including 2 minutes costs $0.63.) This fare is only valid for up to (but not including) 40 minutes.

 

Express this information as a function, where represents the amount of money charged and x represents the number of minutes which have elapsed.
 What is the domain for this function?
 What is the range for this function?
 What is ? What does this value represent in the context of the problem?
 Is this function one-to-one? How do you know?

Answer 1

This function relies on a greatest integer function.  I don't have access to the actual symbol here, but I'll use g(x) to mean the greatest integer ≤ x.  (You should look it up in your text so you can use the symbol).

 

The function would be: f(x) = 0.39 + g(x-1).  (I used x-1 since we charge a different rate for the first minute.)

 

The domain is all the possible times you could enter up to (but not including) 40 minutes.   Therefore it would be {x|0<x<40} or (0,40) depending on which notation you'd use.

 

The range would be a list of all the possible charges, starting at 0.39 and jumping by 0.24 for every additional minute up to  but not including 40 minutes.  Therefore the range would be {0.39, 0.63, 0.87, ..., 9.75}

 

You didn't give enough context for the next part about ? so I'll skip forward to the last question.  This function is not one-to-one, since each element in the range doesn't connect to only one element in the domain.  For example, any value from 1≤x<2 would connect to $0.63 in the range.

Answer 2

This is a step function;  the general equation for a step function is:

 

  f(X) = a⌊ bx - h ⌋ + k

 

a  determines the distance between two consecutive sections

   of the graph

 

b will determine the length of one of the segments

 

h will move the graph right if pos. and left if neg.

 

k will move the graph up if pos. and down if neg.

 

All of these are transformations of the parent graph

(Google and you can see this parent graph)

 

Since our graph does not contain the right endpoint we use

what is called the Floor model of the step function

 

Your equation would be:

 

f(X) = ⌊.24)X ⌋ + .39

(the symbols have endpts. on their bottoms)

 

On your graph, you would have a closed circle at the point (0, .39)

with a segment running to the open circle at the point (1, .39)

 

Then closed circle at (1, .63) with segment to open circle at (2, .63)

 

This pattern would continue until you reached the

open circle at (40 , 9.75)

 

Domain:   0 < X < 40

 

Range:      .39 ≤ Y ≤ $9.75

 

The function is not one to one because there are infinite

X-values assigned to each of the Y-values