1. Combining Functions
Lesson 5.1

2. Functions to Combine
Enter these functions into your calculator

3. Combining Functions Consider the following expressions
Predict what will be the result if you graph

4. 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?

5. 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

6. Population and Supply Viewing the two functions
What is the significance of S(x) – P(x)
What does it look like – graph it

7. Population and Supply
What does it mean?
When should we be concerned?

8. Population and Supply Per capita food supply could be a quotient
When would we be concerned on this formula? Set window -5 < y < 5

9. Combinations Using Tables
Determine the requested combinations x -2 -1 1 2 3
r(x) 5 6 7 8 9
s(x)
s(x)/r(x)
r(x)-s(x)
4 – 2r(x)

10. Assignment A
Lesson 5.1A
Page 346
Exercises 1 – 25 odd, 61, 62

11. Composition of Functions
Value fed to first function
Resulting value fed to second function 
End result taken from second function 

12. Composition of Functions
Notation for composition of functions:
Alternate notation:

13. Try It Out Given two functions: Then p ( q(x) ) =
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) ) 

14. Try It Out q ( p(x) ) = q ( 2x + 1) = (2x + 1)2 – 3 =

15. Using the Calculator
Given
Define these functions on your calculator

16. Using the Calculator Now try the following compositions: g( f(7) )
f( g(3) )
g( f(2) )               
f( g(t) )
g( f(s) )
WHY ??

17. Using the Calculator
Is it also possible to have a composition of the same function?
g( g(3.5) ) = ???

18. Composition Using Graphs
k(x) defined by the graph
j(x) defined by the graph
Do the composition of k( j(x) )

19. 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:

20. Composition Using Graphs
Results of k( j(x) )

21. Composition With Tables
Consider the following tables of values:  x 1 2 3 4 7
f(x)
g(x)
f(g(x)
f(g(1))
g(f(x)
g(f(3))

22. Assignment B
Lesson 5.1B
Page 347
Exercises 27 – 77 EOO