1. Warm – up #6
1. 3𝑥+1 − 𝑥+1 =2 3𝑥+1 = 2+ 𝑥+1 3x + 1 = (2+ 𝑥+1 )(2+ 𝑥+1 ) 3x + 1 = 𝑥+1 + (x + 1) 3x + 1 = x 𝑥+1 2x – 4 = 4 𝑥+1 2x - 4 = 4 𝑥+1
( ) ( ) 2
( ) 2 ( ) 2

2. Warm – up #6 Solutions
2𝑥−4 2𝑥−4 =16 𝑥+1 4𝑥 2 −16𝑥+16=16𝑥+16 4 𝑥 2 −32𝑥=0 4x(x – 8) = 0 4x = 0 x – 8 = 0 x = 0 x = 8

3. Solving Radical Equations
CHECK x = 0
3(0)+1 − 0+1 =2
1 −1=2
CHECK x = 8
3(8)+1 − 8+1 =2
25 − 9 =2
5−3=2

4. Homework Log Wed 1/13 Lesson 6 – 6 Learning Objective:
To perform function operations
Hw: Pg. 401 #9 – 26 all

5. 1/13/16 Lesson 6 – 6 Function Operations Day 1
Algebra II

6. Learning Objective To add functions To subtract functions
To multiply functions
To divide functions

7. Function Operations
𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔 𝑥 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔(𝑥) 𝑓∙𝑔 𝑥 =𝑓 𝑥 ∙𝑔(𝑥) 𝑓 𝑔 𝑥 = 𝑓(𝑥) 𝑔(𝑥) , 𝑔 𝑥 ≠0 Domain needs to be in both 𝑓 & 𝑔

8. Domain
Domain: Look for any restrictions on x (y must be real) x under radical  ≥0 x in denominator of a fraction  ≠0

9. Find 𝑓+𝑔, 𝑓−𝑔, & domain
1. 𝑓 𝑥 =4𝑥+7 𝑔 𝑥 = 𝑥 +𝑥 Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  𝑥≥0 Domain of 𝑓 𝑥 & g 𝑥  𝑥≥0 𝑓+𝑔= 4𝑥+7 +( 𝑥 +𝑥) = 5𝑥+ 𝑥 +7 𝑓−𝑔= 4𝑥+7 −( 𝑥 +𝑥) = 4𝑥+7− 𝑥 −𝑥 = 3𝑥− 𝑥 +7

10. Find 𝑓+𝑔, 𝑓−𝑔, & domain
2. 𝑓 𝑥 = 2𝑥 2 +8 𝑔 𝑥 =𝑥−3 Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  All Real #s Domain of 𝑓 𝑥 & g 𝑥  All Real #s 𝑓+𝑔= 2𝑥 (𝑥−3) = 2𝑥 2 +𝑥+5 𝑓−𝑔= 2𝑥 2 +8 −(𝑥−3) = 2𝑥 2 +8−𝑥+3 = 2𝑥 2 −𝑥+11

11. Find 𝑓∙𝑔 , 𝑓 𝑔 , & domain
3. 𝑓 𝑥 = 𝑥 2 −9 𝑔 𝑥 =𝑥+3 Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  All Real #s Domain of 𝑓 𝑥 & g 𝑥  All Real #s 𝑓∙𝑔= 𝑥 2 −9 (𝑥+3) = 𝑥 3 +3 𝑥 2 −9𝑥−27
𝑥 2 -9 x
𝑥 3
–9x 3
3 𝑥 2
−27

12. Find 𝑓∙𝑔 , 𝑓 𝑔 , & domain
3. 𝑓 𝑥 = 𝑥 2 −9 𝑔 𝑥 =𝑥+3 𝑓 𝑔 = 𝑥 2 −9 𝑥+3 Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  x + 3 ≠ 0  𝑥≠−3 Domain of 𝑓 𝑥 & g 𝑥  𝑥≠−3
= (𝑥+3)(𝑥−3) 𝑥+3
= 𝑥−3

13. Find 𝑓∙𝑔 , 𝑓 𝑔 , & domain
4. 𝑓 𝑥 =3 𝑥 2 −11𝑥−4 𝑔 𝑥 =3𝑥+1 Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  All Real #s Domain of 𝑓 𝑥 & g 𝑥  All Real #s 𝑓∙𝑔= 3 𝑥 2 −11𝑥−4 (3𝑥+1) = 9𝑥 3 −30 𝑥 2 −23𝑥−4
–11x
3𝑥 2 –4 3x
9𝑥 3
-33 𝑥 2
–12x
3 𝑥 2
-11x -4 1

14. Find 𝑓∙𝑔 , 𝑓 𝑔 , & domain = (3𝑥+1)(𝑥−4) 3𝑥+1 = 𝑥−4 –12 –12 1 -11
4. 𝑓 𝑥 =3 𝑥 2 −11𝑥−4 𝑔 𝑥 =3𝑥+1 𝑓 𝑔 = 3𝑥 2 −11𝑥−4 3𝑥+1 3 𝑥 2 −12𝑥+1𝑥−4 3x(x – 4) + 1(x – 4) (3x + 1)(x – 4) Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  3x + 1 ≠ 0  𝑥≠− 1 3 Domain of 𝑓 𝑥 & g 𝑥  𝑥≠− 1 3
= (3𝑥+1)(𝑥−4) 3𝑥+1
= 𝑥−4
–12
–12 1
-11

15. Find 𝑓∙𝑔 , 𝑓 𝑔 , & domain
5. 𝑓 𝑥 =2𝑥+5 𝑔 𝑥 = 𝑥 2 −3𝑥+2 Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  All Real #s Domain of 𝑓 𝑥 & g 𝑥  All Real #s 𝑓∙𝑔= 𝑥 2 −3𝑥+2 (2𝑥+5) = 2𝑥 3 − 𝑥 2 −11𝑥+10
–3x
𝑥 2 2 2x
2𝑥 3
-6 𝑥 2 4x
5 𝑥 2
-15x 10 5

16. Find 𝑓∙𝑔 , 𝑓 𝑔 , & domain = 2𝑥+5 (𝑥−1)(𝑥−2) 2 -2 -1 -3
5. 𝑓 𝑥 =2𝑥+5 𝑔 𝑥 = 𝑥 2 −3𝑥+2 𝑓 𝑔 = 2𝑥+5 𝑥 2 −3𝑥+2 𝑥 2 −2𝑥−1𝑥+2 x(x – 2) – 1(x – 2) (x – 1)(x – 2) Domain of 𝑓 𝑥  All Real #s Domain of g 𝑥  x - 1 ≠ 0 & x - 2 ≠ 0 Domain of 𝑓 𝑥 & g 𝑥  𝑥≠1, 2
= 2𝑥+5 (𝑥−1)(𝑥−2) 2 -2 -1 -3

17. Ticket Out the Door
Perform the following operations for 𝑓 𝑥 = 𝑥 2 −16 𝑔 𝑥 =𝑥−4
𝑓+𝑔
𝑓−𝑔
𝑓∙𝑔
𝑓 𝑔

18. Assignment: Pg. 401 #9 – 26 all