Revisiting the Function

1

• Revisiting the Function
• at the Shopping Junction
• Yojana Sharma

2

• 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.

3
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)
• 
  
• 
  
• 
  
• 
  
  

Frosting
Cake
Milk
Cereal
Fabric Softener
Laundry Detergent
4
If you are a fussy shopper do you lose the
function?


Tea
Honey
Cereal
Lemon
Milk
Raisins
Laundry detergent
Fabric Softener
Bleach
5
Is this a function? If yes, is it 1-1?

• Set X Rule Set Y

Milk
Cereal
Coffee
Pasta
Pasta Sauce
Peanut Butter
Bread
Jelly
6
Standard Teaching Concept

• Think of the function as a machine that receives
  an input and throws out an output.
  f
• Input x Output y
  or f(x)

A
B
7
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
8
.

• 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)

9

• 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.

10

• 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.

11
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

12

• f(x) x2 f is 1-1
  function
• f(x) or -vx f is not a
  function

1
1
2
4
3
9
1
1
-1
4
2
-2
9
3
-3
13

• f(x) x2 f is a
  function
• but is
  not 1-1
• It is
  onto function
• since
  range is the
• entire
  set.

-1
1
1
4
-2
2
-3
9
3
14
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?

15
Composition of functions continued

• Common mistake is to add 251540 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 15of 75 11.25. So
  you would actually pay 75-11.25 63.75!

16
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)

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• Domain of g Range of g Add.
• (original price) Sale price 25 15
• off original price
  off sale
• 
  price

f(g(x))
g(x)
x
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• 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.

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• 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))

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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

21

• 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.

22

• x This adds an
• additional
  filter.
• g(x)
• f(g(x))
• (fog)(x) f(g(x))

23
Example 1

• f(x) x1, 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 x0 is filtered out.
• Domain of fog is all real numbers except x 0.

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Example 2

• f(x) 2/(x1), g(x) 1/x
• fog(x) f(g(x)) f(1/x) 2/(1/x 1)2x/(1x)
• xo 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.

25

• Therefore domain of fog is all real numbers
  except 0, -1.
• Example 3
• f(x) v(x-3), g(x) 2-3x
• Find fog and its domain.

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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.

27
Shopping example of a piecewise defined function
that is discontinuous

• Lets 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 !

28

• 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?

29

• C(x) 10x if 0 lt x 50
• C(x) 9x if 50 lt x 100
• C(x) 8x if x gt 100
• y
• Piecewise
  discontinuous
• function
• x

0
50
100
150
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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?

31

• 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
  .

32

• I am done with shopping for groceries and I am
  standing at the supermarket checkout. A scanner
  records prices of the foods I bought.

33
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.

34

• 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

35

• 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.

36
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.

37
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.