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					Revisiting the Function
at the Shopping Junction

Yojana Sharma
• Function -----------idea of dependence.
• A picnic or a barbecue is a function of the
  weather.
• My presenting at AMATYC is a function of
  the college approving the funds for my
  travel and other expenses.
                          Shopping


• A function is a relationship between two
  quantities. These could be food items or
  household items.
• Set X        Rule          Set Y    (1-1 function)
•
     Cake                    Frosting
    Cereal                    Milk

    Laundry                  Fabric
   Detergent                Softener
 If you are a fussy shopper do you lose the
                  function?



  Tea               Honey
                    Lemon
 Cereal              Milk
                    Raisins
 Laundry             Fabric
detergent           Softener
                     Bleach
      Is this a function? If yes, is it 1-1?

•   Set X            Rule         Set Y


       Cereal                       Milk
       Coffee
    Pasta Sauce                    Pasta
    Peanut Butter
                                   Bread
        Jelly
        Standard Teaching Concept

• Think of the function as a machine that
  receives an input and throws out an
  output.             f

        A                         B




     Input x                Output y or f(x)
  This helps to distinguish between x (the
  argument) , f(function) and f(x) (output).

• But it does not clearly distinguish between
  B and f(A), the image of f or explain the
  concept of onto and 1-1 function.
• Think of the function as a bow.
 quiver bow          target

     A
                       .

• Each object in A is represented by an
  arrow in the quiver.
• The function f is the bow.
• It “shoots” the arrow x into the target B ,
  hitting the spot f(x).
• The collection of all spots hit by an arrow
  from A is called the image of f, f(A)
• If every spot on the target is hit by an
  arrow, f is onto function.
• If no spot gets hit by more than one arrow,
  f is 1-1 function.
Assumption: Archer never misses the target
  and arrows dissolve after impact, so it is
  possible for many arrows to hit the same
  spot.
• The standard definitions of relation and
  function can now be introduced.
• Relation: correspondence between two
  sets; first set is Domain, second set is
  Range; members of the set are called
  elements.
• Function: a relation where each element of
  the first set corresponds to exactly one
  element in the second set.
 Concept of relation as a set of
        ordered pairs
• (cereal, milk), (coffee, milk), (pasta sauce,
  pasta), (cheese , pasta), (peanut butter,
  bread), (jelly, bread).
• This is a function but not 1-1.
• Now replace food and household items
  with numbers.
• Make up shopping example
    f(x) = x2         f is 1-1 function
1               1
2               4
3               9



    f(x) = + or -√x    f is not a function
                 1
1               -1
4                2
9               -2
                 3
                -3
     f(x) = x2     f is a function
-1
                 1
 1                      but is not 1-1
-2               4
 2                      It is onto function
-3               9
 3                      since range is the
                         entire set.
     Composition of functions
• Garments section:
  Sales rack of clothes:

 A skirt costing $100 is on discount at 25%
 and under clearance you are asked to take
 off an additional 15% off the sale price.
 How much will you pay?
     Composition of functions
          continued:
• Common mistake is to add
  25%+15%=40% and assume you will pay
  100 - 40 = $60 for the skirt.
• If you know how to do the math you would
  first do 25% of 100 =25 which would give
  you $75 after discount. Then you would do
  15%of 75= 11.25. So you would actually
  pay 75-11.25 = $63.75!
         The “fog function”
What we did in the last slide was
 composition of functions.
It is a function of a function.
One function takes an output(original price
 $100) and maps it to an output(sale price
 $75). Another function takes this output as
 its input(sale price $75) and maps it to an
 output(checkout price $63.75)
•

        x                               f(g(x))
                         g(x)

  Domain of g         Range of g        Add.
• (original price)   Sale price 25% 15%
                     off original price off sale
                                       price
x corresponds to original price of each item
  on rack. Clothes markdown is 25%.
g(x) = 0.75x represents the price after
  markdown.
Because of clearance, an additional 15% off
  this price.
 So f(g(x)) = 0.85g(x), the checkout price for
  that item.
Let x = $100
g(x) = 0.75(100) = $75
f(g(x)) = 0.85(75) = $ 63.75
The textbook definition of composite
  functions is (fog)(x) = f(g(x))
Textbook definition of domain of
              fog
• It is the set of all real numbers x in the
  domain of g such that g(x) is also in the
  domain of f. This definition is hard for
  students to comprehend.
• Think in terms of “filters”
• There are two filters that allow certain
  values of x into the domain.
• The first filter is g(x).If x is not in the
  domain of g, it cannot be in the domain of
   (fog)(x). Out of the values for x that are in
  the domain of g(x) , only some pass
  through because we restrict the output of
  g(x) to values that are allowable as input
  into f.
        x         This adds an
                  additional filter.
               g(x)



                     f(g(x))

(fog)(x) = f(g(x))
                Example 1

• f(x) = x+1, g(x) = 1/x
• fog(x) = f(g(x)) = f(1/x) = 1/x +1
• Domain of g is all real numbers except 0.
  What is not in the domain of g, cannot be
  in the domain of fog. So x=0 is filtered out.
• Domain of fog is all real numbers except x
  = 0.
               Example 2
f(x) = 2/(x+1), g(x) = 1/x
fog(x) = f(g(x)) = f(1/x) = 2/(1/x +1)=2x/(1+x)
  x=o is not in the domain of g and so is
   filtered out. Also x= -1 is in the domain of
   g but it is not in the domain of f. So it is
   filtered out as well because we restrict the
   output of g(x) to values that are allowable
   as input into f and -1 is not allowable.
  Therefore domain of fog is all real numbers
  except 0, -1.
 Example 3:
f(x) = √(x-3), g(x) = 2-3x
 Find fog and its domain.
  Piece-wise defined functions
• Functions defined in terms of pieces.
• Continuous- you can draw the graph of a
  function without picking up the pencil.
• Discontinuous- cannot do the above;
  graph has holes and /or jumps.
     Shopping example of a
 piecewise defined function that
        is discontinuous
• Let’s visit the “T- Shirt Shop” in the
   shopping junction whose slogan reads
 “ Come to the T-Shirt Shop where
    picking out a t-shirt
    requires a lot less effort ! ”
A sorority representative who wants to order
  custom –made T shirts for the sorority is
  given the following deal by the T-shirt
  shop. If she orders 50 or less T-shirts, the
  cost is $10 /shirt, If she orders more than
  50 but less than or equal to100, the cost is
  $9 /shirt. If she orders more than 100, the
  cost is $8/shirt. What is the cost function
  C(x) as a function of the number of T-
  shirts ordered, that is x?
C(x) = $ 10x if 0 < x ≤ 50
C(x) = $ 9x if 50 < x ≤100
C(x) = $ 8x if x > 100
 y
                           Piecewise discontinuous
                           function
                       x
    0   50 100   150
  Application of Inverse Functions

A store employee at the shopping junction
makes $7 per hour and the weekly number
of hours worked per week, x, varies. If the
store withholds 25% of his earnings for
taxes and social security, what function
f(x) expresses his take home pay each
week? Also what does the inverse function
f-1(x) tell you?
f(x) = 5.25 x because $7- 25% of $7 =
  $5.25. Interchanging x and y and solving
  for y gives f-1 (x) = y = x / 5.25
 the inverse function tells you how many
  hours the employee will have to work to
  bring home $ x .
• I am done with shopping for groceries and
  I am standing at the supermarket
  checkout. A scanner records prices of the
  foods I bought.
     Protection of consumers
• Scanning law for Michigan state:
  If there is a discrepancy between the price
  marked on the item and the price recorded
  by the scanner, the consumer is entitled to
  receive 10 times the difference between
  these prices. This amount must be at least
  $1and at most $5. Also the consumer will
  be given the difference between the prices
  in addition to the amount calculated
  above.
For example, if the difference is 5 cents, you
   should get $1( since 10x5 = 50 cents and
   you must get at least $1) + the difference
   of 5 cents. So you should get $1.05.
If the difference is 25 cents, then 10x25 =
   $2.50 cents, so you would get 2.50 + 0.25
   =$ 2.75
  Inquiry Problem:
a) What is the lowest possible refund?
b) Suppose x is the difference between the
 price scanned and the price marked on the
 item and y is the amount refunded to the
 customer, write a formula for y in terms of
 x.
         Problem continued
c) What would the difference between the
  price scanned and the price marked have
  to be in order to obtain a $ 9.00 refund?
d) Graph y as a function of x.
 To function or not to function?
      That is the question!
“shopkeeper mathematics” was the
  important focus from 1930s to 1950s”
The Comprehensive School mathematics
  Program (1975) advocated that functions
  be used as the main avenue through
  which variables and algebra are
  introduced.
The function concept is the fundamental
  concept of algebra.

				
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posted:3/4/2010
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