Compositions of Functions

1. Section 5-2
Composition of Functions

2. Essential Questions
• How do you perform compositions of functions?
• How do you apply compositions of functions?

3. Vocabulary
1. Composition of Functions:

4. Vocabulary
1. Composition of Functions: The result of using
the results of one function to evaluate the
next function

5. Vocabulary
1. Composition of Functions: The result of using
the results of one function to evaluate the
next function
[f ! g](x ) = f [g(x )]

6. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[f !g](2)a.

7. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[f !g](2)
= f [g(2)]
a.

8. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[f !g](2)
= f [g(2)]
= f [2(2)2
+ 2]
a.

9. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[f !g](2)
= f [g(2)]
= f [2(2)2
+ 2]
= f (10)
a.

10. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[f !g](2)
= f [g(2)]
= f [2(2)2
+ 2]
= f (10)
= −10 + 8
a.

11. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[f !g](2)
= f [g(2)]
= f [2(2)2
+ 2]
= f (10)
= −10 + 8
= −2
a.

12. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[f !g](2)
= f [g(2)]
= f [2(2)2
+ 2]
= f (10)
= −10 + 8
= −2
[f !g](2) = −2
a.

13. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[g !f ](2)b.

14. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[g !f ](2)
= g[f (2)]
b.

15. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[g !f ](2)
= g[f (2)]
= g[−2 + 8]
b.

16. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[g !f ](2)
= g[f (2)]
= g[−2 + 8]
= g(6)
b.

17. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[g !f ](2)
= g[f (2)]
= g[−2 + 8]
= g(6)
= 2(6)2
+ 6
b.

18. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[g !f ](2)
= g[f (2)]
= g[−2 + 8]
= g(6)
= 2(6)2
+ 6
= 78
b.

19. Given and ,
find each value.
Example 1
g(x ) = 2x 2
+ xf (x ) = −x + 8
[g !f ](2)
= g[f (2)]
= g[−2 + 8]
= g(6)
= 2(6)2
+ 6
= 78
[g !f ](2) = 78
b.

20. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.

21. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x )

22. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]

23. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)]

24. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0)

25. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1

26. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)]

27. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7)

28. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7

29. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)]

30. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9)

31. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9) = 4

32. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9) = 4
f [g(8)]

33. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9) = 4
f [g(8)] = f (2)

34. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9) = 4
f [g(8)] = f (2) = 6

35. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9) = 4
f [g(8)] = f (2) = 6
f !g = {(7,−1),(−1,7),(4,4),(8,6)}

36. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9) = 4
f [g(8)] = f (2) = 6
f !g = {(7,−1),(−1,7),(4,4),(8,6)}
D = {−1,4,7,8}

37. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[f !g](x ) = f [g(x )]
f [g(7)] = f (0) = −1
f [g(−1)] = f (7) = 7
f [g(4)] = f (9) = 4
f [g(8)] = f (2) = 6
f !g = {(7,−1),(−1,7),(4,4),(8,6)}
D = {−1,4,7,8}
R = {−1,4,6,7}

38. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.

39. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x )

40. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]

41. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)]

42. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)

43. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6) Undefined

44. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)]
Undefined

45. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4)
Undefined

46. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
Undefined

47. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)]
Undefined

48. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7)
Undefined

49. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7) = 0
Undefined

50. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7) = 0
g[f (0)]
Undefined

51. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7) = 0
g[f (0)]= g(−1)
Undefined

52. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7) = 0
g[f (0)]= g(−1) = 7
Undefined

53. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7) = 0
g[f (0)]= g(−1) = 7
g !f = {(9,9),(7,0),(0,7)}
Undefined

54. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7) = 0
g[f (0)]= g(−1) = 7
g !f = {(9,9),(7,0),(0,7)}
D = {0,7,9}Undefined

55. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f = {(2,6),(9,4),(7,7),(0,−1)}
g = {(7,0),(−1,7),(4,9),(8,2)}
a.
[g !f ](x ) = g[f (x )]
g[f (2)] = g(6)
g[f (9)] = g(4) = 9
g[f (7)] = g(7) = 0
g[f (0)]= g(−1) = 7
g !f = {(9,9),(7,0),(0,7)}
D = {0,7,9}
R = {0,7,9}
Undefined

56. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.

57. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )

58. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )
= f [g(x )]

59. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )
= f [g(x )]
= f [2x −1]

60. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )
= f [g(x )]
= f [2x −1]
= 3(2x −1)2
− (2x −1)+ 4

61. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )
= f [g(x )]
= f [2x −1]
= 3(2x −1)2
− (2x −1)+ 4
= 3(4x 2
− 4x +1)− 2x +1+ 4

62. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )
= f [g(x )]
= f [2x −1]
= 3(2x −1)2
− (2x −1)+ 4
= 3(4x 2
− 4x +1)− 2x +1+ 4
= 12x 2
−12x + 3 − 2x + 5

63. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )
= f [g(x )]
= f [2x −1]
= 3(2x −1)2
− (2x −1)+ 4
= 3(4x 2
− 4x +1)− 2x +1+ 4
= 12x 2
−12x + 3 − 2x + 5
= 12x 2
−14x + 8

64. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[f !g](x )
= f [g(x )]
= f [2x −1]
= 3(2x −1)2
− (2x −1)+ 4
= 3(4x 2
− 4x +1)− 2x +1+ 4
= 12x 2
−12x + 3 − 2x + 5
= 12x 2
−14x + 8
D = {x | x = !}

65. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.

66. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8

67. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8
y = 2(6x 2
− 7x + 4)

68. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8
y = 2(6x 2
− 7x + 4)
x =
−b
2a

69. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8
y = 2(6x 2
− 7x + 4)
x =
−b
2a
x =
7
2(6)

70. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8
y = 2(6x 2
− 7x + 4)
x =
−b
2a
x =
7
2(6)
x =
7
12

71. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8
y = 2(6x 2
− 7x + 4)
x =
−b
2a
x =
7
2(6)
x =
7
12
y = 12( 7
12 )2
−14( 7
12 )+ 8

72. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8
y = 2(6x 2
− 7x + 4)
x =
−b
2a
x =
7
2(6)
x =
7
12
y = 12( 7
12 )2
−14( 7
12 )+ 8
y =
47
12

73. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 12x 2
−14x + 8
R = y | y ≥
47
12
⎧
⎨
⎩
⎫
⎬
⎭
y = 2(6x 2
− 7x + 4)
x =
−b
2a
x =
7
2(6)
x =
7
12
y = 12( 7
12 )2
−14( 7
12 )+ 8
y =
47
12

74. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.

75. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[g !f ](x )

76. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[g !f ](x )
= g[f (x )]

77. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[g !f ](x )
= g[f (x )]
= g[3x 2
− x + 4]

78. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[g !f ](x )
= g[f (x )]
= g[3x 2
− x + 4]
= 2(3x 2
− x + 4)−1

79. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[g !f ](x )
= g[f (x )]
= g[3x 2
− x + 4]
= 2(3x 2
− x + 4)−1
= 6x 2
− 2x + 8 −1

80. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[g !f ](x )
= g[f (x )]
= g[3x 2
− x + 4]
= 2(3x 2
− x + 4)−1
= 6x 2
− 2x + 8 −1
= 6x 2
− 2x + 7

81. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
[g !f ](x )
= g[f (x )]
= g[3x 2
− x + 4]
= 2(3x 2
− x + 4)−1
= 6x 2
− 2x + 8 −1
= 6x 2
− 2x + 7
D = {x | x = !}

82. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.

83. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 6x 2
− 2x + 7

84. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 6x 2
− 2x + 7
x =
−b
2a

85. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 6x 2
− 2x + 7
x =
−b
2a
x =
2
2(6)

86. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 6x 2
− 2x + 7
x =
−b
2a
x =
2
2(6)
x =
1
6

87. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 6x 2
− 2x + 7
x =
−b
2a
x =
2
2(6)
x =
1
6
y = 6(1
6 )2
− 2(1
6 )+ 7

88. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 6x 2
− 2x + 7
x =
−b
2a
x =
2
2(6)
x =
1
6
y = 6(1
6 )2
− 2(1
6 )+ 7
y =
41
6

89. Example 2
For each pair of functions, find and ,
if they exist. State the domain and range for each
combined function.
[f !g](x ) [g !f ](x )
f (x ) = 3x 2
− x + 4
g(x ) = 2x −1
b.
y = 6x 2
− 2x + 7
R = y | y ≥
41
6
⎧
⎨
⎩
⎫
⎬
⎭
x =
−b
2a
x =
2
2(6)
x =
1
6
y = 6(1
6 )2
− 2(1
6 )+ 7
y =
41
6

90. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.

91. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:

92. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)

93. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)
0.96(1400)

94. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)
0.96(1400)
$1344

95. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)
0.96(1400)
$1344
After:

96. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)
0.96(1400)
$1344
After:
0.96(1500)−100

97. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)
0.96(1400)
$1344
After:
0.96(1500)−100
1440 −100

98. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)
0.96(1400)
$1344
After:
0.96(1500)−100
1440 −100
$1340

99. Example 3
Matt Mitarnowski has $100 deducted from
every paycheck to save for retirement. He can
have this deduction taken before state taxes are
applied, which reduces his taxable income. His
state income tax rate is 4%. If Matt earns $1500
every pay period, find the difference in his net
income if he has the retirement deduction taken
before or after state taxes.
Before:
0.96(1500 −100)
0.96(1400)
$1344
After:
0.96(1500)−100
1440 −100
$1340
Matt’s net pay is $4
more if he takes
out his deduction
before taxes.