# Combining Functions

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```					Combining Functions
Lesson 5.1
Functions to Combine
   Enter these functions into your calculator
f ( x)  x  7
2

g ( x)  0.5  2   x

Combining Functions
   Consider the following expressions
f ( x)  g ( x)
f ( x)  g ( x)
g ( x)  f ( x)
f ( x)
 f ( x) 
2
g ( x)
g ( x)
   Predict what will be the result if you graph
Combining Functions
   Turn off the two
original functions (F4)
   Use them in the
expression for the
combined function

   How does this
differ from a
parabola?
Application
   Given two functions having to do with population
P(x) is the number P ( x)  200  (1.025)
x

of people
   S(x) is the number of people who can be supplied
with resources such as food, utilities, etc.
S ( x)  500  5.75 x
   Graph these two functions
   Window at 0 < x < 100 and 0 < y < 1000
Population and Supply
   Viewing the two
functions
   Population
   Supply

   What is the significance of S(x) – P(x)
   What does it look like – graph it
Population and Supply
   What does it mean?
   When should we be concerned?

S ( x)  P( x)
Population and Supply
   Per capita food supply could be a quotient
S ( x)
P( x)
   When would we be concerned on this formula?
Set window
-5 < y < 5
Combinations Using Tables
   Determine the requested combinations

x        -2   -1   0    1     2     3
r(x)      5    5    6    7     8     9
s(x)      -2   2    -2   2    -2     2
s(x)/r(x)
r(x)-s(x)
4 – 2r(x)
Assignment A

   Lesson 5.1A
   Page 346
   Exercises 1 – 25 odd, 61, 62
Composition of Functions

   Value fed to first function
   Resulting value fed to
second function
   End result taken from
second function
Composition of Functions
   Notation for composition of functions:

y  f ( g ( x))
   Alternate notation:

y  f g ( x)
Try It Out
   Given two functions:
   p(x) = 2x + 1
   q(x) = x2 - 3
   Then p ( q(x) ) =
   p (x2 - 3) =
   2 (x2 - 3) + 1 =
   2x2 - 5
   Try determining q ( p(x) )
Try It Out

   q ( p(x) ) =
   q ( 2x + 1) =
    (2x + 1)2 – 3 =
     4x2 + 4x + 1 – 3 =
     4x2 + 4x - 2
Using the Calculator
1
   Given     f ( x)  2  x   g ( x)  2
x

   Define these functions on your calculator
Using the Calculator
Now try the following compositions:
 g( f(7) )

 f( g(3) )              WHY ??

 g( f(2) )

 f( g(t) )

 g( f(s) )
Using the Calculator
   Is it also possible to have a composition of the
same function?

   g( g(3.5) ) = ???
Composition Using Graphs
k(x) defined by the graph     j(x) defined by the graph

Do the composition of k( j(x) )
Composition Using Graphs
   It is easier to see what the function is doing if
we look at the values of
k(x), j(x), and then k( j(x) ) in tables:
Composition Using Graphs
   Results of k( j(x) )
Composition With Tables
    Consider the following tables of values:
x         1      2       3        4        7

f(x)       3      1       4        2        7

g(x)           7      2       1        4        3

f(g(x)     f(g(1))

g(f(x)                      g(f(3))
Assignment B

   Lesson 5.1B
   Page 347
   Exercises 27 – 77 EOO

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