precalculus honors
posted by annemarie
Generate a function that models the situation. Consider the type of function described—it is not linear!
A shipping company charges $4 for the first pound and $3 for each additional pound or part thereof.
As you try to come up with the equation, consider the following:
How much should you be charged if your shipment weighs 0.7 pounds? $4
How much should you be charged if your shipment weighs 1.0 pounds? $4
How much should you be charged if your shipment weighs 1.3 pounds? $7
How much should you be charged if your shipment weighs 1.8 pounds? $7
How much should you be charged if your shipment weighs 2.0 pounds? $7
How much should you be charged if your shipment weighs 2.1 pounds? $10
Make sure that your function gives the correct answer for each of these.
Can someone pease help, all I could figure out was that it uses linear programming and that its a stepwise function. Thanks :)

MathMate
It is indeed a step function, with each step lasting one pound. Note that the beginning of the step is included in the interval, for example, 2 pounds to 2.9999999 pounds will cost 7$. The end of the step must be excluded to avoid two costs for the same weight. This is also one of the properties of a function.
This property can be verified by the vertical line rule, which says that any vertical line drawn on the domain of the function cannot intersect the function at more than once.
For an illustration of the function, see:
http://img402.imageshack.us/img402/5807/1261088031.png 
annemarie
yes but what is the function?

annemarie
What is the y= function? That's what i don't know.

MathMate
"Generate a function..." means it is one that the student is to make it up. It doesn't usually have a name because it is a custommade function to suit the question, although you could name it anything you want.
The "y" function is the numbers in the yaxis, namely cost of shipping in dollars.
The function has been shown in the graph. Study the graph to make sure you understand how it is made up.
Then try to formulate a mathematical expression for the function. You may have to use an existing function to help you put the function in mathematical terms. The existing function is called floor(x).
Floor(x) is a function defined as the largest integer not exceeding x.
For example,
floor(4)=4
floor(4.2)=4
floor(4.999)=4
and
floor(5)=5
Using this function and some doing some figuring, you should be able to come up with a mathematical function that looks like what is shown in the graph in the link. 
annemarie
Thanks I figured it out :)

annemarie
f(x)=4+3*[[x]]

MathMate
You're almost there. The only problem is that it does not work for integer number of pounds, for example, for weight = 1 pound, the cost is 7 dollars according to your function.
The problem is due to an error on my part. Each step on the graph in the earlier link should have had the interval inclusive on the right, and exclusive on the left, since the price goes up only after the weight exceeds an integer number of pounds.
Here's the corrected link:
http://img696.imageshack.us/img696/8564/1261088031a.png
Consequently the function that should have been used is the ceiling function, which is defined as :
ceiling(x) = smallest integer that is greater than or equal to x.
The required shipping cost function should therefore be
f(x) = 1 + ceiling(x)
where x is the number of pounds.
Check my thinking.
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