1. Section 3.3 Composition of Functions

2. Objectives:
Define the composition of functions and describe the relationship between the dependent and independent variables. Identify special properties of composition, and use these properties to analyze functions. Develop a composition test to determine whether two functions are inverses.

3. Composition of Functions
Another way of combining two functions is to form the composition of one with the other.

4. Composition Value fed to first function
Resulting value fed to second function 
End result taken from second function 

5. Notation for composition of functions:
Alternate notation:
This is read as “f of g of x”

6. Question:
Do you think the composition of two functions is commutative??? Let’s try it out!

7. Now let’s find (g ○ f)(x)
Try It Out
Given two functions:
f(x) = 2x + 1
g(x) = x2 - 3
Find (f ○ g)(x)
=f(g(x))
= f(x2 - 3)
= 2(x2 - 3) + 1
= 2x2 - 5
Now let’s find (g ○ f)(x) 

8. Try It Out Given two functions: f(x) = 2x + 1 g(x) = x2 - 3
Find (g ○ f)(x) 
= g(f(x))
= g(2x + 1)
= (2x + 1)2 – 3
= 4x2 + 4x + 1 – 3
= 4x2 + 4x - 2 

9. Therefore it is not always commutative.
Note
In the last example:
(f ○ g)(x) = 2x2 - 5
(g ○ f)(x) = 4x2 + 4x – 2
Therefore it is not always commutative.
(f ○ g)(x) ≠ (g ○ f)(x)

10. Classwork/Homework:
3.3 Commuting Enrichment Worksheet

11. Composition With Tables
Consider the following tables of values: 
Fill in the rest of the table x 1 2 3 4 7
f(x)
g(x)
f(g(x))
g(f(x))

12. Solutions:
a. 4
b. –22

13. Use the graphs of y = f(x) and y = g(x) to find each of the following compositions.
f(g(3))=
f(-1) = -5
g(f(3))=
g(3) = -1
f(g(0))=
f(2) = 4
g(f(0)) =
g(0) = 2
g(g(3)) =
g(-1) = 3

14. Classwork:
Pg 132
Ex: 1 – 4, 6, 7, 10 – 17

15. Composition of Inverse Functions
There is one situation when the composition of two functions are commutative.
This happens with inverse functions!

16. Composition of Inverse Functions
Consider
f(3) = 27   and   f -1(27) = 3
Find f(f -1(27))
= f(3)
= 27 
Notice that f(f -1(x)) = x

17. Composition of Inverse Functions
Consider
f(3) = 27   and   f -1(27) = 3
Find f -1(f(3))
= f -1(27)
= 3
Notice that f -1(f(x)) = x

18. Let’s find the inverse of f(x) = x³ + 2
Order of operations of f (x): 1) Power of 3
2) Add 2
Order of operations of f -1 :
1) Undo addition, or subtract 2
x – 2
2) Undo power of 3, or take third root.
( x – 2 )1/3
Inverse: f -1 (x) = ( x – 2 )1/3
What is f ( f –1 (x))?

19. f (x) = x3 + 2 and f -1(x) = ( x – 2 )1/3
f ( f -1 (x)) =
= ( x – 2 ) + 2
= x
f ( f –1 (x)) = x
Likewise, it can be shown that f -1( f (x)) = x

20. You can show that two functions are inverses by using this:

21. Verify that f(x) = x - 6 and f -1 (x) = x + 6 are inverses
f (f -1(x)) = f(x + 6) = (x + 6) – 6 = x
f -1(f(x)) = f -1 (x – 6 ) = (x – 6) + 6 = x

24. Classwork:
Relations and Functions Worksheet (Double Sided). (Determining whether the inverse is a function and Verifying Inverses)

25. A pebble is dropped into a calm pond.
Ripples of concentric circles form.

26. The area of the circle is given by the function
The radius (in feet) of the outer ripple is r(t) = 0.6t.
t = time in seconds after the pebble strikes water.
The area of the circle is given by the function
(a) Find and interpret
(b) At what time will the area equal 36π feet?

27. (a) Find and interpret Solution:
r(t) = 0.6t
(a) Find and interpret
Solution:
The composite function represents the area of the outer ripple as a function of time.

28. The count will reach 36π feet when t = 10 seconds.
(b) At what time will the area equal 36π feet?
Solution:
The ripple will reach an area of 36π when
36π = 0.36πt²
36π/0.36π = t²
100 = t²
t = 10
The count will reach 36π feet when t = 10 seconds.

29. What would be the independent and dependent variable?
Independent is r(t)
Dependent is A(r(t))

30. Critical Thinking:
Write h(x) as the composition of two functions f(x) and g(x) for which f(g(x)) = h(x).
h(x) = (x – 3)5
f(x) = x5 and g(x) = x – 3
b) h(x) = 2x + 5
f(x) = 2x and g(x) = x
c) h(x) = (x – 3)3 5
f(x) = x3 and g(x) = x – 3

31. Homework:
Section 3.3 Practice and Apply Worksheet