Composition of Functions Sept. 16 and 17

1. Write the equation of the line 1. Write the equation of the line. Simplify final answer so that it is in the form y=mx+b. m=2, (1, -1) 2. Write the transformation, domain, range, and graph the following equation: 𝑦= 𝑥−2 +3 Bell Ringer:

Correct and score out of 46 Homework!

Composition of Functions Write these functions down on your notes: 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 What does it mean to do the following? f(2) 2. g(1) 3. h(3) Explain in words, then do the process that each task is asking. Compare your answers to your neighbor. Did you come to the same conclusions? Composition of Functions

Composition of Functions So f(2) is asking us to evaluate 2 in the function f(x). f(x)= 2𝑥−1=2 2 −1=4−1=3 This gives you the point (2, 3) g(1) is asking you to evaluate 1 in the function g(x). g(x)= −3𝑥+5=−3 1 +5=−3+5=2 This gives you the point (1, 2) h(3) is asking you to evaluate 3 in the function h(x). h(x)= 𝑥 2 −1= 3 2 −1=9−1=8 This gives you the point (3, 8) Composition of Functions

Composition of Functions 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 Now find: 𝑓 −3 = ℎ −2 = 𝑔 5 = Composition of Functions

Composition of Functions 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 So you know what it means to evaluate a number within a function, what do you think this is telling you? 𝑓 𝑔 𝑥 Discuss with your neighbor for 30 seconds. Go! Composition of Functions

Composition of Functions 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 𝑓(𝑔 𝑥 ) is telling you to evaluate the function 𝑔(𝑥) in the function 𝑓(𝑥). You will replace all 𝑥’s in the 𝑓(𝑥) function with the 𝑔 𝑥 answer then simplify. 𝑓 𝑔 𝑥 =2 −3𝑥+5 −1 Then simplify… −6𝑥+10−1 So 𝑓 𝑔 𝑥 =−6𝑥+9 Composition of Functions

Composition of Functions 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 What is 𝑔 𝑓 𝑥 tell you to do? Describe and then find what it is asking you to do. Composition of Functions

Composition of Functions Are the answers for 𝑓 𝑔 𝑥 and 𝑔 𝑓 𝑥 the same? Composition of Functions

Composition of Functions 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 What does 𝑓 𝑔 1 tell you to do? Is there more than one way to do this process? Composition of Functions

Composition of Functions 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 Option 1: 𝑓(𝑔 1 ) You can put 𝑔(𝑥) into all the 𝑥’s of 𝑓(𝑥) and simplify. Then evaluate for 𝑥=1. 𝑓 𝑔 𝑥 =2 −3𝑥+5 −1 This combines to 𝑓 𝑔 𝑥 =−6𝑥+9 Then we plug 1 in for x. 𝑓 𝑔 1 =−6 1 +9 This equals 3. Composition of Functions

Composition of Functions 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 =−3𝑥+5 ℎ 𝑥 = 𝑥 2 −1 Option 2: 𝑓 𝑔 1 You can also evaluate 𝑔 1 , then take the answer and evaluate it in 𝑓 𝑥 . 𝑔 1 =−3 1 +5=2 Then take that answer and plug into 𝑓 𝑥 . 𝑓 2 =2 2 −1=4−1=3 Composition of Functions

Composition of Functions Composition of Functions can be denoted in two different ways. 𝑓 𝑔 𝑥 vs. 𝑓∘𝑔 𝑥 Both are telling us the same thing. To compose the function of g into the function of f. That is put 𝑔(𝑥) into every 𝑥 of 𝑓. The second guy is going into the first guy listed. Composition of Functions

Composition of Functions Find: 𝑔 𝑓 6 𝑓 𝑔 10 𝑔∘𝑓(7) Composition of Functions

Composition of Functions Given: 𝑓 𝑥 =3𝑥−4 𝑔 𝑥 = 𝑥 2 +1 Find: 𝑓 𝑔 −3 What are you being asked to do? Put -3 into g, then the answer into f. Or compose the function g into f, then evaluate at x=-3. Composition of Functions

Composition of Functions Given: 𝑓 𝑥 =3𝑥−4 𝑔 𝑥 = 𝑥 2 +1 Find: 𝑔 𝑓 0 Composition of Functions

Given: 𝑓 𝑥 =3𝑥−4 𝑔 𝑥 = 𝑥 2 +1 Find: f(a)

Given: 𝑓 𝑥 =3𝑥−4 𝑔 𝑥 = 𝑥 2 +1 Find: 𝑓 𝑔 3

Given: 𝑓 𝑥 =3𝑥−4 𝑔 𝑥 = 𝑥 2 +1 Find: 𝑓∘𝑔(𝑥)

Given: 𝑓 𝑥 =3𝑥−4 𝑔 𝑥 = 𝑥 2 +1 Find: 𝑔 𝑎+1

Given: 𝑓 𝑥 =3𝑥−4 𝑔 𝑥 = 𝑥 2 +1 Find: 𝑔 𝑓 𝑚

Worksheet!