1. Bellringer August 27th
Write a function whose graph is 𝑦= 𝑥 3 but is shifted up 3 units and left 1 unit.
Identify the shift from the parent function for 𝑓 𝑥 =− 1−𝑥 −1
Identify 5 points on an xy table for the parent function of 𝑓 𝑥 = 𝑥 2 with the indicated shifts
𝑦=2𝑓(𝑥) and 𝑦=𝑓(2𝑥)

2. 3.4 Operations on functions and Composition of Functions
MAFS.912.F-BF.1.1

3. Consider the following functions: 𝑓 𝑥 = 𝑥 2 +2𝑥−3 and 𝑔 𝑥 =𝑥+1
1. Find 𝑓 𝑥 +𝑔(𝑥)= ____________________
(𝑓+𝑔)(𝑥)= ________________ this is the _________________
2. Find 𝑓 𝑥 −𝑔(𝑥)= _____________________ (𝑓−𝑔)(𝑥)= ________________ this is the _________________
𝑥 2 +2𝑥−3 +𝑥+1
𝑥 2 +3𝑥−2
Sum function
𝑥 2 +2𝑥−3 −(𝑥+1)
𝑥 2 +𝑥−4
Difference function

4. Consider the following functions: 𝑓 𝑥 = 𝑥 2 +2𝑥−3 and 𝑔 𝑥 =𝑥+1
3. Find 𝑓 𝑥 ∙𝑔(𝑥)= _________________________ (𝑓∙𝑔)(𝑥)= ________________ this is the _________________ 4. Find 𝑓(𝑥) 𝑔(𝑥) = _______________________________ Identify any restrictions ( 𝑓 𝑔 )(𝑥)= ________________ this is the _________________
(𝑥 2 +2𝑥−3 )(𝑥+1)
𝑥 3 +3 𝑥 2 −𝑥−3
Product function
𝑥 2 +2𝑥−3 𝑥+1
𝑥 2 +2𝑥−3 𝑥+1 , 𝑥≠−1
Quotient function

5. At a glance 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔(𝑥) 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔(𝑥) 𝑓∙𝑔 𝑥 =𝑓 𝑥 ∙𝑔(𝑥)
Function
Notation
Domain
Sum
{domain of f}∩{domain of g}
Difference
Product
Quotient
{domain of f}∩{domain of g}∩{𝑔(𝑥)≠0}
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)

6. Operations on Functions: Determining Domains of New Functions
Consider the following functions then find the sum, difference, product, and quotient functions and their domain.
𝑓 𝑥 = 𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 = 4−𝑥
Sum:____________________
Difference:_______________
Product:_________________
Quotient:________________
This intersection is the domain for ____________________ functions.
The _________________ function requires an additional restriction of ________.
Therefore the domain is ________________
𝑥−1 + 4−𝑥
[1, ∞)
Domain of 𝑓 𝑥 =___________
Domain of 𝑔 𝑥 =____________
The intersection of these domains is:
________________
𝑥−1 − 4−𝑥
(−∞,4]
(𝑥−1)(4−𝑥)
𝑥−1 4−𝑥
[1,4]
Sum, difference, product
quotient
𝑥≠4
[1,4)

7. Finding a composite function and its domain
Consider the functions 𝑓 𝑥 = 𝑥 2 +1 𝑎𝑛𝑑 𝑔 𝑥 =𝑥−3
find 𝑓∘𝑔 𝑥 .
Write 𝑓 𝑥 using “place holder” notation ____________________
Express the composite function 𝑓∘𝑔 ____________________
Substitute 𝑔 𝑥 =𝑥−3 into f ____________________
Eliminate the parenthesis ____________________
𝑓 ___ = [__] 2 +1
𝑓 𝑔(𝑥) = [𝑔(𝑥)] 2 +1
𝑓 𝑔(𝑥) = [𝑥−3] 2 +1
𝑓 𝑔(𝑥) = 𝑥 2 −6𝑥+10

8. The domain of the composite function is the _________ of the ___________ of each original function. (Reminder: check for _________________ on your new function) Example: find the domain of 𝑓∘𝑔 𝑓∘𝑔 𝑥 𝑖𝑓 𝑓 𝑥 = 1 𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 = 1 𝑥
union
domains
restrictions
D of 𝑓 𝑥 = −∞, 1 ∪ 1,∞ 𝐷 𝑜𝑓 𝑔 𝑥 =(−∞,0)∪(0,∞)
D of 𝑓∘𝑔 𝑥 =(−∞,0)∪ 0,1 ∪(1,∞)

9. Evaluating a Composite function
Given the functions 𝑓 𝑥 = 𝑥 2 −7 𝑎𝑛𝑑 𝑔 𝑥 =5− 𝑥 2 find the following a. 𝑓(𝑔 1 ) b. 𝑓(𝑔 −2) c. 𝑔(𝑓 3 )
𝑓(𝑔 1 )=9
𝑓 𝑔 −2 =−6
𝑓 𝑔 3 =1

10. Homework Due August 31st
TB p even