Advanced Functions and Equations Part

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					Advanced Functions and
      Equations
        Part I
Composition and operations on functions,
           Transformations
           Sections 3.4, 3.5
Advanced Functions and
      Equations
 Composition and operations on functions
              Section 3.5
Operations of functions
expl:
A certain company has two factories. The first factory has a
  revenue of R1(x) = 3x + 40 where x is the number of
  units made. The second factory has a revenue of
  R2(x) = 4x + 50.

1.) What is the total revenue of both factories?
2.) How much more does factory 2 make?
For part 1, you need to find the sum of the revenues.
  Algebraically, let’s say
  R1 ( x)  R2 ( x)
   3x  40  4 x  50
   7 x  90
 For part 2, you need to find the difference of the revenues.
   Algebraically, let’s say
 R2 ( x)  R1 ( x)                         Put parentheses
                                           around 3x + 40
                
  4 x  50  3x  40                     because you’re
                                             subtracting.
  4 x  50  3x  40
  x  10
Notation
The following notation is often used.

       ( f  g )( x)  f ( x)  g ( x)
       ( f  g )( x)  f ( x)  g ( x)
       ( f  g )( x)  f ( x)  g ( x)
       f        f ( x)
        ( x ) 
       g               , g ( x)  0
                g ( x)
Domains: all real numbers in both domains of f and g
and, in the case of (f/g)(x), exclude those numbers that
make g(x) = 0
expl: #5, pg 306
Let    f ( x)  x
       g ( x)  3x  5          Remember the
                             domain will be those
Find    a.) f + g            numbers that satisfy
        b.) f – g               both functions.
                              In the case of the
        c.) fg               quotient, we exclude
        d.) f / g            those x’s that would
        and their domains.     cause division by
                                     zero.
  a.)   f ( x)  g ( x)         b.)   f ( x)  g ( x)
         x  3x  5                   x  (3 x  5)
                                       x  3x  5
  domain:     x0               domain:    x0


c.)   f ( x)  g ( x)     d.)    f ( x)     x
                                        
       x 3x  5               g ( x) 3 x  5
                                                   5
                          domain:       x  0, x 
domain:    x0                                     3
Composition
expl:
A popular umbrella manufacturer has this information.
  P(x) = .0025x2 + .2x + 4, where P(x) represents the
  monthly profit of the company and x represents the
  number of umbrellas sold that month, x  0
  U(t) = 80t + 20, where U(t) represents the number of
  umbrellas sold per month and t represents the number of
  days it rains that month, 0  t  31
We are interested in how profit is related to the number of
  days of rain. Let’s find the equation.
A specific example
 Say we have had 12 days of rain this month. Find the
   number of umbrellas we can expect to sell and the profit
   this should yield.

U (12)  80(12)  20  980               If it rains 12
                                        days, we can
                                        expect to sell
                                               980
  …Now find             P(980)          umbrellas…
 the profit we
  make if we             .0025(980)  .2(980)  4
                                        2

   sell 980
  umbrellas.             .0025  960400  196  4
                         2601 If we sell 980
                                    umbrellas, we will
                                    make a profit of $2601.
                       Number of
Number of
                       umbrellas
days of rain
                         U(t)


                       Number of   Profit
                       umbrellas   P(x)


  We’d like to bypass the
   number of umbrellas
     stage. We’d like a
   formula that gives us
  profit given number of
        days of rain.
P( x)  .0025x 2  .2 x  4
  …where x represents the number of umbrellas.
  But U(t) also represents the number of umbrellas.
  So we can substitute U(t) in for x to find P(U(t)).

                                       Apply the rule of
 P(U (t ))                           P(x) to the number
 P(80t  20)                               80t + 20

 .002580t  20  .280t  20  4
                    2


                                
 .0025 6400t 2  3200t  400  .280t  20  4
 16t 2  8t  1  16t  4  4
 16t 2  24t  9
So P (t )  16 t  24 t  9 where t is the number of
                2

  days of rain and P(t) is the profit.

We used composition of functions to find P(U(t)). Now we
 know the relationship between profit and number of days
 of rain.

Check the specific example. If it rains 12 days, what’s the
  profit?
          P(12 )  16 (12 )  24 (12 )  9  2601
                          2



Composition notation:
     ( f  g )(x)  f ( g ( x))
expl: #14 ac, pg 306
Let   f ( x)  3x  2              Remember f(x) and
      g ( x)  2 x  1
                    2              g(x) are rules. They
                                    tell you what to do
a.) Find    f  g 4 .             to x, no matter
c.) Find    f  f 1 .            what that value is
                                           called.

Verify results (will do part a on grapher).
For part a,
     f  g 4
     f ( g ( 4))                 g(4) = 2(4)2 – 1 = 31

     f (31)
     3(31)  2                      For part c,

     95                                     f  f 1
                                             f ( f (1))
              f(1) = 3(1) + 2 =              f (5)
                      5
                                             3(5)  2
                                             17
Verify part a on grapher
   f  g x                First, find the
   f ( g ( x ))              equation for
                              y   f  g x 
   f (2 x  1)
            2


   3( 2 x  1)  2
            2


   6x  3  2
        2


   6x2 1

Then graph it and use the calculator to calculate the y
  value when x is 4.
                                On the Graph
                               screen, press 2nd
                              TRACE (CALC) to
                                   get to the
                               Calculate menu.
                              Then select Value.
                              Give it an input of 4
                              for x, it will tell you
                                   what y is.

In order for the Value or          EVAL on the
EVAL function to work, your         TI-85 or 86
x value must be within the
window dimensions.
Domain of f ( g ( x))
  all real numbers x that are in the domain of g and such
  that g(x) is in the domain of f.


expl: Find f(g(x)) and determine its domain.
         f ( x)  x
         g ( x)  3 x  2                      First find
                                                f(g(x)).
                    f ( g ( x))
                     f (3x  2)
                     3x  2
    The domain of this composite
  function are those numbers that
would work in g(x) = 3x + 2, all real
  numbers, and those g(x) values
that would work in f. Now f requires
   that its inputs are not negative.
                  So…



                         Domain:   3x  2  0
                                       2
                                   x
                                        3
You could also think
  of this as simply
finding the domain,
those numbers that
     work, of the
          3x 
   y function2




                        Look at its graph!
                                    2
                       Domain:   x
                                     3
expl: # 52
Show that f(g(x)) = x and g(f(x)) = x for

f ( x)  4  3 x

g ( x)  4  x 
        1                          We will see
                                  later how this
        3                        proves the two
                                  functions are
                                     inverses.
Show f(g(x)) = x.                   Show g(f(x)) = x.

                       Apply the
  f ( g ( x))          function’s
                                      g ( f ( x))
     1                  rule         g (4  3x)
  f  4  x 
     3         
                                       4  ( 4  3 x) 
                                       1
        1                            3
  4  3 4  x 
                                       4  4  3 x 
        3                            1
  4  4  x                         3
                  Simplify.
                                       3 x 
                                       1
  44 x         Watch the
                     order of          3
 x
                    operations!       x
Worksheet
“Functions: Composition and operations” provides practice
  in performing operations and compositions as well as
  determining domain.




3.5 homework: 1, 9, 13, 27, 32, 33, 47, 51, 55, 65, 67, 77
Advanced Functions and
      Equations
       Transformations
         Section 3.4
Worksheets
“Transformations” introduces several transformations. Most
   of the functions we will deal with this semester are
   transformations of the functions discussed in “Library of
   functions”. These functions, such as y = x 2, y = x, and
   others, are called mother functions.

“Part II” of this worksheet summarizes the transformations
  for which you will be responsible.

“Transformations 2” and “Library of functions and
   transformations 2” provide more practice.
Intro to Transformations worksheet
Part I:
Graph carefully. Check your values with your neighbors often
   to make sure you are on the right track.
1. Complete the table and graph f ( x)  x and h( x)  x 2  3 .
                                           2




        x          -3    -2     -1     0      1      2      3

   f ( x)  x 2

 h( x )  x  3
            2
       x         -3   -2   -1   0   1   2   3

  f ( x)  x 2    9   4    1    0   1   4   9

h( x)  x 2  3 12    7    4    3   4   7   12



  In the table,
    each h(x)                                    Notice how
    value is 3                                   for each x,
 more than the                                   the point is
 correspondin                                     moved up
   g f(x) value                                   3 units to
 because we                                      make h(x).
   added 3 to
    each f(x).
Transformations worksheet
 Let c be a positive    Transformation of     Number of question
   real number.              graph             which is example


c  f ( x), c  1       Vertical stretch by
                                                    5
                        factor of c

                        Vert. compression
c  f ( x), 0  c  1   by factor of c              6

                        Horizontal shift
     f ( x  c)         to right c units            4

                        Horizontal shift
     f ( x  c)         to left c units             3
Transformations worksheet cont.
 Let c be a positive   Transformation of   Number of question
   real number.             graph           which is example


    f ( x)  c         Vertical shift
                       up c units                1


    f ( x)  c         Vertical shift
                                                 2
                       down c units


     f (x)            Reflection about          7
                       x axis


     f (  x)          Reflection about          8
                       y axis
You can think through transformations by imagining what is
  happening to the y values.
For instance, if you compare
   y1  x and y 2  5x , you’ll notice the second
          2                  2

  function’s y values are merely five times the first
  function’s y values. So each point is stretched away from
  the x-axis. The graph will be elongated. So we say the
  graph is vertically stretched by a factor of 5. Check it out
  by graphing both on your calculator!



3.4 homework: 1 – 12, 13 – 16, 17, 19, 21, 23, 27, 33, 46,
   53, 63, 69

				
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