1.06 - Composite Functions

University:
High School
Course:
High School Physics
Academic year:

2024
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411

Pages:

13
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1.06 Composite Functions 𝑓∘𝑔 Reads: The composite of 𝑔 and 𝑓 The Idea: Take 𝑔(𝑥) and substitute it in for the 𝑥 of 𝑓(𝑥) In words… 𝑓∘𝑔 = 𝑓∘𝑔 𝑥 =𝑓 𝑥 ∘𝑔 𝑥 =𝑓 𝑔 𝑥 Notation • If 𝑓 𝑥 = 3 𝑥 2 + 8 and 𝑔 𝑥 = 6𝑥 • Find: 𝑔∘𝑓 𝑔∘𝑓 𝑥 =𝑔 𝑓 𝑥 3 =𝑔 𝑥+8 2 3 =6 𝑥+8 2 = 9𝑥 + 48 Example 1 • If 𝑓 𝑥 = 𝑥 2 − 2𝑥 and 𝑔 𝑥 = 2𝑥 + 1 • Find: 𝑓∘𝑔 𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥 = 𝑓 2𝑥 + 1 = 2𝑥 + 1 2 − 2 2𝑥 + 1 = 4𝑥 2 + 4𝑥 + 1 − 4𝑥 − 2 = 4𝑥 2 − 1 Example 2 • If 𝑓 𝑥 = 𝑥 2 − 2𝑥 and 𝑔 𝑥 = 2𝑥 + 1 • Find: (𝑓 ∘ 𝑔)(2) 𝑓∘𝑔 𝑥 =𝑓 𝑔 2 = 𝑓 2𝑥 + 1 = 2𝑥 + 1 2 − 2 2𝑥 + 1 = 4𝑥 2 + 4𝑥 + 1 − 4𝑥 − 2 = 4𝑥 2 − 1 = 4(2)2 −1 = 15 Example 3 • OR (𝑓 ∘ 𝑔)(2) 𝑓∘𝑔 𝑥 =𝑓 𝑔 2 =𝑓 2 2 +1 =𝑓 5 𝑓 5 = (5)2 −2 5 = 15 • If 𝑓 𝑥 = 𝑥 2 − 2𝑥 and 𝑔 𝑥 = 2𝑥 + 1 • Find: 𝑔∘𝑓 𝑔∘𝑓 𝑥 =𝑔 𝑓 𝑥 = 𝑔 𝑥 2 − 2𝑥 = 2 𝑥 2 − 2𝑥 + 1 = 2𝑥 2 − 4𝑥 + 1 Example 4 • If 𝑓 𝑥 = 𝑥 2 − 2𝑥 and 𝑔 𝑥 = 2𝑥 + 1 • Find: 𝑔∘𝑔 𝑔∘𝑔 𝑥 =𝑔 𝑔 𝑥 = 𝑔 2𝑥 + 1 = 2 2𝑥 + 1 + 1 = 4𝑥 + 2 + 1 = 4𝑥 + 3 Example 5 • If 𝑓 𝑥 = 6𝑥 − 4 • Find: 𝑓 ∘ 𝑓 −1 𝑓 −1 𝑥 ⇒ 𝑥 = 6𝑦 − 4 𝑥 + 4 = 6𝑦 1 𝑥+4 =𝑦 6 Example 7 𝑓 ∘ 𝑓 −1 𝑥 = 𝑓 𝑓 −1 𝑥 1 =𝑓 𝑥+4 6 1 =6 𝑥+4 −4 6 =𝑥+4−4 =𝑥 • If 𝑓 𝑥 = 8 𝑥 − 1 + 3 • Find: 𝑓 ∘ 𝑓 −1 𝑓 −1 𝑥 ⇒ 𝑥 = 8 𝑦 − 1 + 3 𝑥−3 = 𝑦−1 8 𝑥−3 2 +1=𝑦 64 Example 8 𝑓 ∘ 𝑓 −1 𝑥 = 𝑓 𝑓 −1 𝑥 1 =𝑓 𝑥−3 2+1 64 =8 1 𝑥−3 64 2 +1 −1+3 1 =8 𝑥−3 64 2 +3 1 =8 𝑥−3 +3 8 =𝑥−3+3 =𝑥 • If 𝑓 𝑥 = 8 𝑥−4 + 10 • Find: 𝑓 −1 ∘ 𝑓 𝑓 −1 8 𝑥 ⇒ +4 𝑥 − 10 Example 9 𝑓 −1 ∘ 𝑓 𝑥 = 𝑓 −1 𝑓 𝑥 8 −1 =𝑓 + 10 𝑥−4 8 = +4 8 + 10 − 10 𝑥−4 8 = +4 8 𝑥−4 =𝑥−4+4 =𝑥 •𝑓 ∘ −1 𝑓 =𝑥 ALWAYS!!! −1 •𝑓 ∘𝑓 =𝑥 •Visions 1 •P26: 4f,g •P68: 7e Homework