Example 17 - Show that f(x) = 3x+4/5x-7, g(x) = 7x+4/5x-3 - Composite funcions

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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
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Example 17 Show that if f : R – 7﷮5﷯﷯ → R – 3﷮5﷯﷯ is defined by f(x) = 3𝑥 + 4﷮5𝑥 − 7﷯ and g: R − 3﷮5﷯﷯→ R – 7﷮5﷯﷯ is defined by g(x) = 7𝑥 + 4﷮5𝑥 − 3﷯, then fog = IA and gof = IB, where A = R − 3﷮5﷯﷯ , B = R – 7﷮5﷯﷯ ; IA (x) = x, ∀ x ∈ A, IB (x) = x, ∀ x ∈ B are called identity functions on sets A and B, respectively. f(x) = 3𝑥 + 4﷮5𝑥 − 7﷯ & g(x) = 7𝑥 + 4﷮5𝑥 − 3﷯ Finding gof g(x) = 7𝑥 + 4﷮5𝑥 − 3﷯ g(f(x)) = 7𝑓(𝑥) + 4﷮5𝑓(𝑥) − 3﷯ gof = 7 3𝑥 + 4﷮5𝑥 − 7﷯﷯ + 4﷮5 3𝑥 + 4﷯﷮ 5𝑥 − 7﷯﷯﷯ − 3﷯ = 7 3𝑥 + 4﷯ + 4(5𝑥 − 7)﷮5𝑥 − 7﷯﷮ 5 3𝑥 + 4﷯ −3(5𝑥 − 7)﷮5𝑥 − 7﷯﷯ = 7 3𝑥 + 4﷯ + 4(5𝑥 − 7)﷮5 3𝑥 + 4﷯ −3(5𝑥 − 7)﷯ = 21𝑥 + 28 + 20𝑥 − 28﷮15𝑥 + 20 − 15𝑥 + 21﷯ = 41𝑥﷮41﷯ = x Thus, gof = x = IB Finding fog f(x) = 3𝑥 + 4﷮5𝑥 − 7﷯ f(g(x)) = 3𝑔(𝑥) + 4﷮5𝑔(𝑥) − 7﷯ = 3 7𝑥 + 4﷮5𝑥 − 3﷯﷯ + 4﷮5 7𝑥 + 4﷯﷮ 5𝑥 − 3﷯﷯﷯ − 3﷯ = 3 7𝑥 + 4﷯ + 4(5𝑥 − 3)﷮5𝑥 − 3﷯﷮ 5 7𝑥 + 4﷯ − 3(5𝑥 − 3)﷮5𝑥 − 3﷯﷯ = 3 7𝑥 + 4﷯ + 4(5𝑥 − 3)﷮5 7𝑥 + 4﷯ − 3(5𝑥 − 3)﷯ = 21𝑥 + 12 + 20𝑥 − 12﷮35𝑥 + 20 − 35𝑥 + 21﷯ = 41𝑥﷮41﷯ = x Thus, fog = x = IA

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