Composition of Functions

1. Chapter 8: Inverses
and Radicals

2. Section 8-1
Composition of Functions

3. What are composite functions?

4. Activity p. 478
Choose a sticker price OVER $10,000. How much will
you pay if:
A. The rebate is given first, and then the discount?
B. The discount is given first, and then the rebate?
VOTE

5. Example 1
3
r(x) = x − 500 and d(x) = x 4
a. Find the formula for d(r(x))

6. Example 1
3
r(x) = x − 500 and d(x) = x 4
a. Find the formula for d(r(x))
d(r(x)) = d(

7. Example 1
3
r(x) = x − 500 and d(x) = x 4
a. Find the formula for d(r(x))
d(r(x)) = d( x − 500)

8. Example 1
3
r(x) = x − 500 and d(x) = x 4
a. Find the formula for d(r(x))
3
d(r(x)) = d( x − 500) = (x − 500)
4

9. Example 1
3
r(x) = x − 500 and d(x) = x 4
a. Find the formula for d(r(x))
3 3
d(r(x)) = d( x − 500) = (x − 500) = x − 375
4 4

10. Example 1
3
r(x) = x − 500 and d(x) = x 4
a. Find the formula for d(r(x))
3 3
d(r(x)) = d( x − 500) = (x − 500) = x − 375
4 4
3
d(r(x)) = x − 375
4

11. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.

12. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500))

13. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500)

14. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500) = d(4000)

15. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500) = d(4000)
3
= (4000)
4

16. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500) = d(4000)
3
= (4000) = 3000
4

17. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500) = d(4000)
3
= (4000) = 3000
4
OR

18. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500) = d(4000)
3
= (4000) = 3000
4
OR
3
d(r(4500)) = (4500) − 375
4

19. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500) = d(4000)
3
= (4000) = 3000
4
OR
d(r(4500)) = (4500) − 375 = 3375 − 375
3
4

20. Example 1
3
r(x) = x − 500 and d(x) = x 4
b. Evaluate d(r(x)) for x = 4500.
d(r(4500)) = d(4500 − 500) = d(4000)
3
= (4000) = 3000
4
OR
d(r(4500)) = (4500) − 375 = 3375 − 375
3
4
= 3000

21. Composite
gof

22. Composite
gof
The function that maps x onto g( f (x)), whose domain is
the set of all values in the domain of f, then also in g.

23. Composite
gof
The function that maps x onto g( f (x)), whose domain is
the set of all values in the domain of f, then also in g.
g o f (x) = g( f (x))

24. Composite
gof
The function that maps x onto g( f (x)), whose domain is
the set of all values in the domain of f, then also in g.
g o f (x) = g( f (x))

25. Composite
gof
The function that maps x onto g( f (x)), whose domain is
the set of all values in the domain of f, then also in g.
g o f (x) = g( f (x)) g of f of x; Do f first, then g

26. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))

27. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) 5

28. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) 5
1
= 2( ) −1
5

29. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) 5
1
= 2( ) −1
5
2
= −1
5

30. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) 5
1
= 2( ) −1
5
2
= −1
5
3
=− 5

31. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) = g(2(5) −1)
5
1
= 2( ) −1
5
2
= −1
5
3
=− 5

32. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) = g(2(5) −1)
5
1
= 2( ) −1 = g(10 −1)
5
2
= −1
5
3
=− 5

33. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) = g(2(5) −1)
5
1
= 2( ) −1 = g(10 −1)
5
= g(9)
2
= −1
5
3
=− 5

34. Example 2
1
f (x) = 2x −1, g(x) = x
a. f ( g(5)) b. g( f (5))
1
= f( ) = g(2(5) −1)
5
1
= 2( ) −1 = g(10 −1)
5
= g(9)
2
= −1
5
1
3
=
=− 9
5

35. Example 3
2
p(x) = x , q(x) = −x +1
a. p oq(x)

36. Example 3
2
p(x) = x , q(x) = −x +1
a. p oq(x)
= p(q(x))

37. Example 3
2
p(x) = x , q(x) = −x +1
a. p oq(x)
= p(q(x))
= p(−x +1)

38. Example 3
2
p(x) = x , q(x) = −x +1
a. p oq(x)
= p(q(x))
= p(−x +1)
2
= (−x +1)

39. Example 3
2
p(x) = x , q(x) = −x +1
a. p oq(x)
= p(q(x))
= p(−x +1)
2
= (−x +1)
2
= x − 2x +1

40. Example 3
2
p(x) = x , q(x) = −x +1
b. q o p(x)

41. Example 3
2
p(x) = x , q(x) = −x +1
b. q o p(x)
= q( p(x))

42. Example 3
2
p(x) = x , q(x) = −x +1
b. q o p(x)
= q( p(x))
2
= q(x )

43. Example 3
2
p(x) = x , q(x) = −x +1
b. q o p(x)
= q( p(x))
2
= q(x )
2
= −(x ) +1

44. Example 3
2
p(x) = x , q(x) = −x +1
b. q o p(x)
= q( p(x))
2
= q(x )
2
= −(x ) +1
2
= −x +1

45. Example 3
2
p(x) = x , q(x) = −x +1
c. q oq(x)

46. Example 3
2
p(x) = x , q(x) = −x +1
c. q oq(x)
= q(q(x))

47. Example 3
2
p(x) = x , q(x) = −x +1
c. q oq(x)
= q(q(x))
= q(−x +1)

48. Example 3
2
p(x) = x , q(x) = −x +1
c. q oq(x)
= q(q(x))
= q(−x +1)
= −(−x +1) +1

49. Example 3
2
p(x) = x , q(x) = −x +1
c. q oq(x)
= q(q(x))
= q(−x +1)
= −(−x +1) +1
= x −1 +1

50. Example 3
2
p(x) = x , q(x) = −x +1
c. q oq(x)
= q(q(x))
= q(−x +1)
= −(−x +1) +1
= x −1 +1
=x

51. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of

52. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f:

53. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers

54. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
g( f (x))

55. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
g( f (x)) = g(x + 2)

56. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
1
g( f (x)) = g(x + 2) =
2(x + 2) −1

57. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
1 1
g( f (x)) = g(x + 2) = =
2(x + 2) −1 2x + 4 −1

58. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
1 1 1
g( f (x)) = g(x + 2) = = =
2(x + 2) −1 2x + 4 −1 2x + 3

59. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
1 1 1
g( f (x)) = g(x + 2) = = =
2(x + 2) −1 2x + 4 −1 2x + 3
2x + 3 ≠ 0

60. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
1 1 1
g( f (x)) = g(x + 2) = = =
2(x + 2) −1 2x + 4 −1 2x + 3
2x + 3 ≠ 0 2x ≠ −3

61. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
1 1 1
g( f (x)) = g(x + 2) = = =
2(x + 2) −1 2x + 4 −1 2x + 3
3
x≠−
2x + 3 ≠ 0 2x ≠ −3 2

62. Example 4
1
, f (x) = x + 2
g(x) =
2x −1
g o f (x).
Find the domain of
Domain of f: All real numbers
1 1 1
g( f (x)) = g(x + 2) = = =
2(x + 2) −1 2x + 4 −1 2x + 3
3
x≠−
2x + 3 ≠ 0 2x ≠ −3 2
3
D = {x : x ≠ − } 2

63. Homework

64. Homework
p. 481 #1-23, skip 21
“Anyone who takes himself too seriously always runs the
risk of looking ridiculous; anyone who can consistently
laugh at himself does not.” - Vaclav Havel