Perform the indicated operations to combine the given

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Perform the indicated operations to combine the given functions. Express the final answer in its simplest form. Given: ?(?) = 3? 2 − ? − 4 , ?(?) = 3? + 3 Find: a. ?(?) − ?(?) b. ?(?)[2?(?)] c. ?(?) ?(?) d. (? ∘ ?)(?) Let’s do something interesting ! How can the two functions ?(?) = ? 2 − 4 and ?(?) = ? 2 + 4 be combined so that the resulting function is equal to −8? How about when the resulting function is equal to 2? 2 ? Do you have an idea about how to do it? Actually, when you subtract ? 2 + 4 from ? 2 − 4 it will result in -8. So combining functions ? and ? by subtraction represented as (? − ?)(?) = ?(?) − ?(?) = (? 2 − 4) − (? 2 + 4) remember the rule in subtraction = ? 2 − 4 − ? 2 − 4 change the sign of the subtrahend (? − ?)(?) = −8 Second question, please. Yes! To answer the next question, combine the two functions by addition! Adding the two functions, (? + ?)(?) = ?(?) + ?(?) = (? 2 − 4) + (? 2 + 4) = ? 2 − 4 + ? 2 + 4 (? − ?)(?) = 2? 2
Two or more functions can be combined to form new functions, A way of combining functions could be an application of one or more of the fundamentals operations, addition, and multiplication or to perform addition, subtraction, multiplication, or division. Example 1: Given: ?(?) = 5? 2 − 17? + 6 , ?(?) = 2 − 5? Find: a. ? − ? c. ? ? b. ?. ? d. 5? − 2? Solution: a. . ? − ? = ?(?) − ?(?) = (5? 2 − 17? + 6) − (2 − 5?) = 5? 2 − 17? + 6 − 2 + 5? change the sign of the subtrahend ? − ? = 5? 2 − 12? + 4 b. ?? = ?(?)?(?) = (5? 2 − 17? + 6)(2 − 5?) distribution = 10? 2 − 34? + 12 − 25? 3 + 75? 2 − 30? = −25? 3 + 85? 2 − 64? + 12 c. ? ? = ?(?) ?(?) = 5? 2 −17?+6 2−5? = (5?−2)(?−3) 2−5? = (5?−2)(?−3) −(5?−2) = −? + 3 To divide division algorithm can be applied instead of factoring d. 5? − 2? = 5?(?) − 2?(?) = 5(2 − 5?) − 2(5? 2 − 17? + 6) multiply ? ?? 5 and ? ?? 2 = 10 − 25? − 10? 2 + 34? − 12 then, combine similar terms = −10? 2 + 9? − 2 Operations on Functions Let functions ? and ? 1. Sum of Functions , ? + ? is defined as (? + ?)(?) = ?(?) + ?(?) 2. Difference of Functions , , ? − ? is defined as (? − ?)(?) = ?(?) − ?(?) 3. Product of Functions , ?? is defined as (??)(?) = ?(?)?(?) 4. Quotient of Functions , ?/? is defined as (?/?)(?) = ?(?) ?(?)
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10 Example 2: Given: ?(?) = 1 ?+1 , ?(?) = ?+1 ?−1 Find: a. ?(2) + ?(2?) ?. ?(−2?) ?(−2?) Solution: a. There’s a need to evaluate the two functions before adding. ?(2) + ?(2?) = 1 2 + 1 + 2? + 1 2? − 1 ?(2) ?(2?) = 1 3 + 2?+1 2?−1 add the fractions = 2?−1+3(2?+1) 3(2?−1) get the LCD = 2?−1+6?+3 6?−3 ?(2) + ?(2?) = 8?+2 6?−3 b. 𝐻(−2?) ?(−2?) = 1 −2? +1 −2?+1 −2?−1 = 1 −2?+1 −2?−1 −2?+1 get the reciprocal = −2? − 1 (−2? + 1)(−2? + 1) = − 2?+1 (1−2?) 2 Another way of combining functions to form a new function is by composition. This is a procedure of forming a new function by substituting a function into another function. If there are two functions ? and ? , another function ℎ(?) = ?(?(?)) can be computed by substituting ? into ? . It is a composition of ? and ? , denoted by ? ∘ ? ???? ?? " ? ?????? ?" or ′??????????? ?? ? ??? ?" Definition of Composite Function The composite function ? ∘ ? of functions ? and ? is defined by (? ∘ ?)(?) = ?(?(?))
11 Example 3: Given: ?(?) = √? , ?(?) = √? 2 − 9 Find: a. ? ∘ ? b. ? ∘ ? c. ? ∘ ? ∘ ? Solution: a. (? ∘ ?)(?) = ?(?(?)) since ?(?) = √? , substitute this to ? = ?( √? ) evaluating the function ? = ( √? ) 2 − 9 = √? − 9 b. (? ∘ ?)(?) = ?(?(?)) this time substitute ?(?) = √? 2 − 9 into ? = ?(√? 2 − 9 ) = ? 2 − 9 = √? 2 − 9 4 c. . ? ∘ ? ∘ ? = ? (?(?(?))) from the inner function ?(?) = ? (?( √? )) then, ?(√? ) = √? − 9 = ?( ? − 9 ) = √? − 9 = √? − 9 4 What would you do? What would you do to answer this problem? Problem: If g (?) = 5 + 3? and ?(?) = 2 + 3? + 3? 2 , find a function such that ? ∘ ℎ = ? .