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Inverse of a function
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Inverse of a function
Last updated at May 29, 2023 by Teachoo
Ex1.3 , 4 If 𝑓(𝑥)=(4𝑥 − 3)6𝑥 − 4, 𝑥 ≠ 23 , show that 𝑓𝑜𝑓(𝑥)=𝑥, for all 𝑥 ≠ 23 . What is the inverse of f? 𝑓(𝑥)=(4𝑥 − 3)6𝑥 − 4 𝑓(𝑓𝑥) = 4𝑓(𝑥) − 36𝑓(𝑥) − 4 𝑓𝑜𝑓𝑥 = 44𝑥 − 36𝑥 − 4 − 364𝑥 − 36𝑥 − 4 − 4 = 44𝑥 − 3 − 36𝑥 − 46𝑥 − 464𝑥 − 3 − 46𝑥 − 46𝑥 − 4 = 16𝑥 − 12 − 18𝑥 +126𝑥 − 424𝑥 − 18 − 24𝑥 +166𝑥 − 4 = 16𝑥 − 12 − 18𝑥 +126𝑥 − 4 × 6𝑥 − 424𝑥 − 18 − 24𝑥 + 16 = 16𝑥 − 12 − 18𝑥 +1224𝑥 −18 −24𝑥 +16 = −2𝑥 + 00 − 2 = −2𝑥− 2 = x ∴ 𝑓𝑜𝑓𝑥 = x Calculating inverse of f(x) 𝑓(𝑥)=(4𝑥 − 3)6𝑥 − 4 Put f(x) = y y = (4𝑥 − 3)6𝑥 − 4 y(6x – 4) = (4x – 3) 6xy – 4y = 4x – 3 6xy – 4x = 4y – 3 x(6y – 4) = 4y – 3 x = 4𝑦 − 36𝑦 − 4 So, inverse of f = 4𝑦 − 36𝑦 − 4 ∴ Let inverse of f = g (y) = 4𝑦 − 36𝑦 − 4 g (y) = 4𝑦 − 36𝑦 − 4 Replacing y with x g (x) = 4𝑥 − 36𝑥 − 4 = f(x) Hence we can say inverse of f is f itself i.e. f -1 = f