Warm – up #6 1. 3𝑥+1 − 𝑥+1 =2 3𝑥+1 = 2+ 𝑥+1 3x + 1 = (2+ 𝑥+1 )(2+ 𝑥+1 ) 3x + 1 = 4 + 4 𝑥+1 + (x + 1) 3x + 1 = x + 5 + 4 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

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

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

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

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

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

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

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

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 2 +8 +(𝑥−3) = 2𝑥 2 +𝑥+5 𝑓−𝑔= 2𝑥 2 +8 −(𝑥−3) = 2𝑥 2 +8−𝑥+3 = 2𝑥 2 −𝑥+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

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

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

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

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

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

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

Assignment: Pg. 401 #9 – 26 all