Section 6.5 Operations with Functions Algebra 2 Section 6.5 Operations with Functions

Warm Up Simplify. Assume that all expressions are defined. #1 2𝑥+5 −( 𝑥 2 +3𝑥−2) #2 (𝑥−1) 𝑥+1 2 #3 𝑥 2 −𝑥−6 𝑥 2 −4

Operations on functions is very much like operations on expressions. You’re creating a new function!

Addition 𝒇+𝒈 𝒙=𝒇 𝒙 +𝒈(𝒙) Example 𝑓 𝑥 =2 𝑥 2 −5𝑥+11 𝑔 𝑥 =5 𝑥 2 +2𝑥−30 𝑓+𝑔 𝑥=2 𝑥 2 −5𝑥+11+5 𝑥 2 +2𝑥−30 𝑓+𝑔 𝑥=7 𝑥 2 −3𝑥−19 Would 𝑔+𝑓 𝑥 be a different function?

Subtraction 𝒇−𝒈 𝒙=𝒇 𝒙 −𝒈(𝒙) 𝑓 𝑥 =4𝑥−2 𝑔 𝑥 =6𝑥+8 𝑓−𝑔 𝑥= 4𝑥−2 − 6𝑥+8 𝑓−𝑔 𝑥=4𝑥−2−6𝑥−8 𝑓−𝑔 𝑥=−2𝑥−10 Would 𝑔−𝑓 𝑥 be a different function?

Try these Pg. 446 #2-4

Multiplication 𝒇𝒈 𝒙=𝒇(𝒙)∙𝒈(𝒙) 𝑓 𝑥 =𝑥−3 𝑔 𝑥 =4𝑥+2 𝑓𝑔 𝑥= 𝑥−3 4𝑥+2 𝑓𝑔 𝑥= 𝑥−3 4𝑥+2 𝑓𝑔 𝑥= 4𝑥 2 +2𝑥−12𝑥−6 𝑓𝑔 𝑥=4 𝑥 2 −10𝑥−6 Would 𝑔𝑓 𝑥 be a different function?

Division 𝒇 𝒈 x= 𝒇(𝒙) 𝒈(𝒙) 𝒈(𝒙)≠𝟎 𝑓 𝑥 = 𝑥 2 +7𝑥+12 𝑔 𝑥 =𝑥+3 𝑓 𝑔 𝑥= 𝑥 2 +7𝑥+12 𝑥+3 𝑓 𝑔 𝑥= (𝑥+4)(𝑥+3) (𝑥+3) 𝑓 𝑔 𝑥=𝑥+4

Same or Different? Would 𝑓 𝑔 𝑥 and 𝑔 𝑓 𝑥 be the same function?

Try these! Pg. 446 #5-7

Composition of functions Composition of functions- this operation uses the output of one function as the input of another function 𝒇𝒐𝒈 𝒙=𝒇 𝒈 𝒙 This operation is NOT commutative!!! 𝒈𝒐𝒇 𝒙=𝒈(𝒇 𝒙 )

Start with the innermost parentheses 1st!

Try These! Pg. 8-10

Boots are 50% off Take an additional 40% off Are you getting 90% off Boots are 50% off Take an additional 40% off Are you getting 90% off? When you get 40% off, what percent of the price are you paying?

𝑔 𝑥 =.50𝑥 𝑓 𝑥 =.40𝑥 𝑓𝑜𝑔 𝑥=𝑓 𝑔 𝑥 𝑓 .50𝑥 =.60 .50𝑥 What percent are you getting off of the original price?

𝑓 𝑥 = 2𝑥 2 𝑔 𝑥 =𝑥+2 Find 𝑓𝑜𝑔 𝑥= ? 𝑔𝑜𝑓 𝑥= ?

Try these! Pg. 446 #11-14

Homework/Classwork Pg. 446 #15-33 all, 45-47 all