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)

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

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

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

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)

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)

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

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,∞)

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

Homework Due August 31st TB p 286 2-52 even