combining functions lesson 5.1. functions to combine enter these functions into your calculator

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

Assign the Composition of FunctionsGeogebra WorksheetDue in 1 Week