Operations with Functions

1. Section 5-1
Operations with Functions

2. Essential Questions
• How do you perform arithmetic operations
with functions?
• How do you apply arithmetic operations with
functions?

3. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division

4. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )

5. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)

6. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f + g)(x ) = x + 5

7. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = f (x )− g(x )
(f + g)(x ) = x + 5

8. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = f (x )− g(x )
(f − g)(x ) = 2x − (−x + 5)
(f + g)(x ) = x + 5

9. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = 3x − 5
(f − g)(x ) = f (x )− g(x )
(f − g)(x ) = 2x − (−x + 5)
(f + g)(x ) = x + 5

10. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = 3x − 5
(f − g)(x ) = f (x )− g(x )
(f − g)(x ) = 2x − (−x + 5)
(f + g)(x ) = x + 5
(f i g)(x ) = f (x )i g(x )

11. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = 3x − 5
(f − g)(x ) = f (x )− g(x )
(f − g)(x ) = 2x − (−x + 5)
(f + g)(x ) = x + 5
(f i g)(x ) = f (x )i g(x )
(f i g)(x ) = 2x(−x + 5)

12. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = 3x − 5
(f − g)(x ) = f (x )− g(x )
(f − g)(x ) = 2x − (−x + 5)
(f + g)(x ) = x + 5
(f i g)(x ) = f (x )i g(x )
(f i g)(x ) = −2x 2
+10x
(f i g)(x ) = 2x(−x + 5)

13. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = 3x − 5
(f − g)(x ) = f (x )− g(x )
(f − g)(x ) = 2x − (−x + 5)
(f + g)(x ) = x + 5
(f i g)(x ) = f (x )i g(x )
(f i g)(x ) = −2x 2
+10x
(f i g)(x ) = 2x(−x + 5)
f
g
⎛
⎝⎜
⎞
⎠⎟ (x ) =
f (x )
g(x )
,g(x ) ≠ 0

14. Operation Definition
Example
Let f(x) = 2x and g(x) = −x + 5
Addition
Subtraction
Multiplication
Division
(f + g)(x ) = f (x )+ g(x )
(f + g)(x ) = 2x + (−x + 5)
(f − g)(x ) = 3x − 5
(f − g)(x ) = f (x )− g(x )
(f − g)(x ) = 2x − (−x + 5)
(f + g)(x ) = x + 5
(f i g)(x ) = f (x )i g(x )
(f i g)(x ) = −2x 2
+10x
(f i g)(x ) = 2x(−x + 5)
f
g
⎛
⎝⎜
⎞
⎠⎟ (x ) =
f (x )
g(x )
,g(x ) ≠ 0
f
g
⎛
⎝⎜
⎞
⎠⎟ (x ) =
2x
−x + 5
,x ≠ 5

15. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.

16. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.
= f (x )+ g(x )

17. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.
= f (x )+ g(x )
= 3x 2
+ 7x + 2x 2
− x −1

18. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.
= f (x )+ g(x )
= 3x 2
+ 7x + 2x 2
− x −1
= 5x 2
+ 6x −1

19. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.
= f (x )+ g(x )
= 3x 2
+ 7x + 2x 2
− x −1
= 5x 2
+ 6x −1
= f (x )− g(x )

20. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.
= f (x )+ g(x )
= 3x 2
+ 7x + 2x 2
− x −1
= 5x 2
+ 6x −1
= f (x )− g(x )
= 3x 2
+ 7x − (2x 2
− x −1)

21. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.
= f (x )+ g(x )
= 3x 2
+ 7x + 2x 2
− x −1
= 5x 2
+ 6x −1
= f (x )− g(x )
= 3x 2
+ 7x − (2x 2
− x −1)
= 3x 2
+ 7x − 2x 2
+ x +1

22. Example 1
Given and ,
find each function.
g(x ) = 2x 2
− x −1f (x ) = 3x 2
+ 7x
(f − g)(x )a. (f + g)(x ) b.
= f (x )+ g(x )
= 3x 2
+ 7x + 2x 2
− x −1
= 5x 2
+ 6x −1
= f (x )− g(x )
= 3x 2
+ 7x − (2x 2
− x −1)
= 3x 2
+ 7x − 2x 2
+ x +1
= x 2
+ 8x +1

23. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
a. (f i g)(x )

24. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
a. (f i g)(x )
= f (x )i g(x )

25. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
a. (f i g)(x )
= f (x )i g(x )
= (3x 2
− 2x +1)(x − 4)

26. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
a. (f i g)(x )
= f (x )i g(x )
= (3x 2
− 2x +1)(x − 4)
= 3x 3
−12x 2
− 2x 2
+ 8x + x − 4

27. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
a. (f i g)(x )
= f (x )i g(x )
= (3x 2
− 2x +1)(x − 4)
= 3x 3
−12x 2
− 2x 2
+ 8x + x − 4
= 3x 3
−14x 2
+ 9x − 4

28. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
b.
f
g
⎛
⎝⎜
⎞
⎠⎟ (x )

29. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
b.
f
g
⎛
⎝⎜
⎞
⎠⎟ (x )
=
f (x )
g(x )

30. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
b.
f
g
⎛
⎝⎜
⎞
⎠⎟ (x )
=
f (x )
g(x )
=
3x 2
− 2x +1
x − 4

31. Example 2
Given and ,
find each function. Indicate any restrictions
in the domain or range.
g(x ) = x − 4f (x ) = 3x 2
− 2x +1
b.
f
g
⎛
⎝⎜
⎞
⎠⎟ (x )
=
f (x )
g(x )
=
3x 2
− 2x +1
x − 4
; x ≠ 4

32. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
a. Write a function C(x) that represents the total
cost of the tickets, where x is the number of
students on the trip.

33. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
a. Write a function C(x) that represents the total
cost of the tickets, where x is the number of
students on the trip.
C(x ) = 3(24)+15x

34. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
a. Write a function C(x) that represents the total
cost of the tickets, where x is the number of
students on the trip.
C(x ) = 3(24)+15x
C(x ) = 15x + 72

35. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
b. Everyone who goes on the trip will split the
total cost of the tickets evenly. Write a function
N(x) to represent the number of people who will
contribute.

36. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
b. Everyone who goes on the trip will split the
total cost of the tickets evenly. Write a function
N(x) to represent the number of people who will
contribute.
N(x ) = x + 3

37. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
c. Find and explain what this function
represents.
C
N
⎛
⎝⎜
⎞
⎠⎟ (x )

38. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
c. Find and explain what this function
represents.
C
N
⎛
⎝⎜
⎞
⎠⎟ (x )
C(x ) = 15x + 72

39. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
c. Find and explain what this function
represents.
N(x ) = x + 3
C
N
⎛
⎝⎜
⎞
⎠⎟ (x )
C(x ) = 15x + 72

40. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
c. Find and explain what this function
represents.
N(x ) = x + 3
C
N
⎛
⎝⎜
⎞
⎠⎟ (x )
C(x ) = 15x + 72
C
N
⎛
⎝⎜
⎞
⎠⎟ (x )

41. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
c. Find and explain what this function
represents.
N(x ) = x + 3
C
N
⎛
⎝⎜
⎞
⎠⎟ (x )
C(x ) = 15x + 72
C
N
⎛
⎝⎜
⎞
⎠⎟ (x ) =
15x + 72
x + 3

42. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
c. Find and explain what this function
represents.
N(x ) = x + 3
C
N
⎛
⎝⎜
⎞
⎠⎟ (x )
C(x ) = 15x + 72
C
N
⎛
⎝⎜
⎞
⎠⎟ (x ) =
15x + 72
x + 3
This represents the dollar
amount each person will pay

43. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
d. If 20 students go on the trip, how much will
each person pay?

44. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
d. If 20 students go on the trip, how much will
each person pay?
C
N
⎛
⎝⎜
⎞
⎠⎟ (20)

45. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
d. If 20 students go on the trip, how much will
each person pay?
C
N
⎛
⎝⎜
⎞
⎠⎟ (20) =
15(20)+ 72
(20)+ 3

46. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
d. If 20 students go on the trip, how much will
each person pay?
C
N
⎛
⎝⎜
⎞
⎠⎟ (20) =
15(20)+ 72
(20)+ 3
=
300 + 72
23

47. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
d. If 20 students go on the trip, how much will
each person pay?
C
N
⎛
⎝⎜
⎞
⎠⎟ (20) =
15(20)+ 72
(20)+ 3
=
300 + 72
23
=
372
23

48. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
d. If 20 students go on the trip, how much will
each person pay?
C
N
⎛
⎝⎜
⎞
⎠⎟ (20) =
15(20)+ 72
(20)+ 3
=
300 + 72
23
=
372
23
= $16.18

49. Example 3
Matt Mitarnowski is buying tickets to an
aquarium for a school trip. Adult tickets cost
$24 and student tickets cost $15. There will be
3 adult chaperones on the trip.
d. If 20 students go on the trip, how much will
each person pay?
C
N
⎛
⎝⎜
⎞
⎠⎟ (20) =
15(20)+ 72
(20)+ 3
=
300 + 72
23
=
372
23
= $16.18
Each person will pay $16.18 to cover the cost of the trip.