1. Warm Up Determine the domain of the function.
f(x) = 7𝑥 −42
7x – 42 ≥ 0
x ≥ 6

2. Operations and Compositions of Functions

3. Operations with Functions Sum Difference Product Quotient
(f + g)(x) =
f(x) + g(x)
(f – g)(x) =
f(x) – g(x)
(f • g)(x) =
f(x) • g(x)
𝑓(𝑥) 𝑔(𝑥)
( 𝑓 𝑔 )(x) =

4. f(x) + g(x) f(x) – g(x) Examples 1. f(x) = 2x + 1 g(x) = x – 2
Find each:
a. (f + g)(x)
b. (f – g)(x)
f(x) + g(x)
(2x + 1) + (x – 2)
3x – 1
f(x) – g(x)
(2x + 1) – (x – 2)
2x + 1 – x + 2
x + 3

5. f(x) ● g(x) (2x + 1) ● (x – 2) 2 𝑥 2 – 4x + x – 2 2 𝑥 2 – 3x – 2
c. (f • g)(x) d. ( 𝑓 𝑔 )(x)
f(x) ● g(x)
(2x + 1) ● (x – 2)
2 𝑥 2 – 4x + x – 2
2 𝑥 2 – 3x – 2
𝑓(𝑥) 𝑔(𝑥)
2𝑥+1 𝑥−2
; x ≠ 2

6. Examples 2. f(x) = 2x + 1 g(x) = x – 2 Find each: a. (f + g)(6)
f(x) + g(x) =
3x – 1
13 + 4
3(6) – 1 17 17

7. b. (f – g)(6)
f(x) – g(x) =
x + 3
f(6) – g(6)
6 + 3
13 – 4 9 9

8. f(x) • g(x) = 2 𝑥 2 – 3x – 2 f(6) • g(6) 2 (6) 2 – 3(6) – 2
c. (f • g)(6)
f(x) • g(x) =
2 𝑥 2 – 3x – 2
f(6) • g(6)
2 (6) 2 – 3(6) – 2
2(36) – 18 – 2
13 • 4
72 – 18 – 2 52 52

9. 𝑓(𝑥) 𝑔(𝑥) 2𝑥+1 𝑥−2 = 𝑓(6) 𝑔(6) = 2(6)+1 6−2 13 4 ; x ≠ 2 = 12+1 6−2
d. 𝑓 𝑔 (6)
2𝑥+1 𝑥−2 =
𝑓(6) 𝑔(6)
= 2(6)+1 6−2
13 4
; x ≠ 2
= −2
= 13 4
; x ≠ 2

10. Examples 3. f(x) = x + 3 Find each: a. (f + g)(x)
f(x) + g(x)
(x + 3) + (2𝑥) (𝑥 − 5)
(𝑥+3) (2𝑥) (𝑥 − 5)
● (𝑥 − 5) (𝑥 − 5) 1
𝑥 2 −15 𝑥 −5
𝑥 2 − 5𝑥 + 3𝑥 −15 𝑥 −5 + (2𝑥) (𝑥 − 5)
; x ≠ 5

11. (𝑥+3) – (2𝑥) (𝑥 − 5) f(x) – g(x) (x + 3) – (2𝑥) (𝑥 − 5)
b. (f – g)(x)
f(x) – g(x)
(x + 3) – (2𝑥) (𝑥 − 5)
(𝑥+3) – (2𝑥) (𝑥 − 5)
● (𝑥 − 5) (𝑥 − 5) 1
𝑥 2 − 5𝑥 + 3𝑥 −15 𝑥 −5 – (2𝑥) (𝑥 − 5)
𝑥 2 − 5𝑥 + 3𝑥 −15 −2𝑥 𝑥 −5
𝑥 2 −4𝑥 −15 𝑥 −5
; x ≠ 5

12. f(x) ● g(x) f(x) = x + 3 (𝑥+ 3) 1 ● (2𝑥) (𝑥 − 5) g(x) = 𝟐𝒙 𝒙 − 𝟓
(𝑥+ 3) 1 ● (2𝑥) (𝑥 − 5)
g(x) = 𝟐𝒙 𝒙 − 𝟓
2 𝑥 2 +6𝑥 𝑥 −5
c. (f • g)(x)
d. ( 𝑓 𝑔 )(x)
; x ≠ 5
(𝑥 + 3) 1 ÷ (2𝑥) (𝑥 −5)
(𝑥+3) (2𝑥) (𝑥 −5)
(𝑥 + 3) 1 ● (𝑥 − 5) (2𝑥)
𝑥 2 −2𝑥 −15 2𝑥
𝑥 2 −5𝑥 +3𝑥 −15 2𝑥
; x ≠ 0, 5

13. Homework: Function Operation WS #1 – 7, 9 – 15, 17, 20

14. Warm Up
Given the graph, state the domain and range and determine whether or not it represents a function:
𝑫𝒐𝒎𝒂𝒊𝒏:−𝟏≤𝒙≤𝟓
𝑹𝒂𝒏𝒈𝒆:−𝟑≤𝒚≤𝟑
Not a function

15. (f о g)(x) = f(g(x)) (g о f)(x) = g(f(x)) Composition of Functions
- the process of combining two functions where one function is performed first and the result of which is substituted in place of each x in the other function.
read as “f of g of x”
read as “g of f of x”
(f о g)(x)
= f(g(x))
(g о f)(x)
= g(f(x))

16. Examples 3. f(x) = 2x + 1 g(x) = x – 2 Find each: a. (f о g)(x)

17. = g(f(x)) = g(2x + 1) = (2x + 1) – 2 = 2x – 1
3. f(x) = 2x + 1 g(x) = x – 2 b. (g о f)(x)
= g(f(x))
= g(2x + 1)
= (2x + 1) – 2
= 2x – 1

18. Examples 3. f(x) = x2 + 4 g(x) = x – 4 a. (f (g(x))
= x2 – 4x – 4x
= x2 – 8x + 20

19. = g(f(x)) = g(x2 + 4) = (x2 + 4) – 4 = x2 b. (g(f(x)) 3. f(x) = x2 + 4

20. Homework: Function Operation WS #8, 16, 18, 19, 21, 22