Composition of Functions
Lesson 8.1
Introduction
• Value fed to first
function
• Resulting value fed to
second function
• End result taken from
second function
Introduction
• 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
• Given
f ( x)  2  x
1
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)
g(f(x)
f(g(1))
g(f(3))
Decomposition of Functions
Someone once dug up Beethoven's tomb
and found him at a table busily erasing
stacks of papers with music writing on
them. They asked him ... "What are you
doing down here in your grave?" He
responded, "I'm de-composing!!"
But, seriously folks ...
Consider the following function which
could be a composition of two different
functions.
2
1
k ( j (t ))     2
t
Decomposition of Functions
• The function could be decomposed into two
functions, k and j
1
j (t ) 
t
2
k (t )  t  2
Assignment
• Lesson 8.1
• Page 359
• Exercises 1 – 59 odd