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)
g ( x)

 f ( x) 
2
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)
of people
S(x) is the number of people who can be supplied
with resources such as food, utilities, etc.
x
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 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:
y  f ( g ( x))

Alternate notation:
y  f g ( x)
Try It Out

Given two functions:



Then p ( q(x) ) =




p(x) = 2x + 1
q(x) = x2 - 3
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
f ( x)  2  x
1
g ( x)  2
x

Given

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)
g(f(x)
f(g(1))
g(f(3))
Assignment B




Lesson 5.1B
Page 380
Exercises 57 - 77 EOO
95, 97
Assign the Composition of Functions
Geogebra Worksheet
Due in 1 Week