Functions 099

Algebra Level 2

If f(f^-1(f(f^-1(f.........f(x))))....)))=3x+2 Then [f(f3.5))]=? Where [.] denotes Greatest integer function

34 39 38 43

Ananth Raviraj Shetty by Ananth Raviraj Shetty , 16, India

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1 solution

Tom Engelsman
Dec 6, 2020

We have a multi-composition of a linear function f ( x ) = A x + B f(x) = Ax + B where each f − 1 ( f ( x ) ) = x f^{-1}(f(x)) =x . Taking the composition all the way to completion leaves us with the original linear function itself, hence f ( x ) = 3 x + 2. f(x) = 3x+2. Calculating ⌊ f ( f ( 3.5 ) ) ⌋ = ⌊ 3 ⋅ [ 3 ( 3.5 ) + 2 ] + 2 ⌋ = ⌊ 39.5 ⌋ = 39 . \lfloor f(f(3.5)) \rfloor = \lfloor 3 \cdot [3(3.5) + 2] + 2 \rfloor = \lfloor 39.5 \rfloor = \boxed{39}.

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