combining functions lesson 5.1. functions to combine enter these functions into your calculator
TRANSCRIPT
Combining Functions
Lesson 5.1
Functions to Combine
Enter these functions into your calculator
2( ) 7
( ) 0.5 2xf x x
g x
Combining Functions
Consider the following expressions
Predict what will be the result if you graph
2
( ) ( )
( ) ( )
( ) ( )
( )( ) ( )
( )
f x g x
f x g x
g x f x
f xf x g x
g x
Combining Functions
Turn off the two original functions (F4)
Use them in theexpression for thecombined function
How does this differ from a parabola?
Application
Given two functions having to do with population P(x) is the number
of people S(x) is the number of people who can be supplied
with resources such as food, utilities, etc.
Graph these two functions Window at 0 < x < 100 and 0 < y < 1000
( ) 200 (1.025)xP x
( ) 500 5.75S x x
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
When would we be concerned on this formula?Set window-5 < y < 5
( )
( )
S x
P x
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 378 Exercises 1 – 37 EOO
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:
Alternate notation:
( ( ))y f g x
( )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
Given
Define these functions on your calculator
2
1( ) 2 ( )f x x g x
x
Using the Calculator
Now try the following compositions: g( f(7) ) f( g(3) ) g( f(2) ) f( g(t) ) g( f(s) )
WHY ??
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 ofk(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 380 Exercises 57 - 77 EOO
95, 97
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