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Let \(f(x)=2x+1\). Find the sum of all \(x\) that satisfy the equation \(f^{-1}(x)=f(x^{-1})\).

 Feb 17, 2019
 #1
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\(f^{-1}(x) = f(x^{-1})\\ f(f^{-1}(x)) = f(f(x^{-1})\\ x = f(f(x^{-1}))\)

 

\(x = 2f(x^{-1})+1 = \\ 2\left(\dfrac{2}{x}+1\right)+1 = \\ \dfrac 4 x + 3\\ x^2 = 4+5x\\ x^2-3x - 4 = 0\\ (x-4)(x+1) = 0\\ x=-1, 4\)

 

\(-1 + 4 = 3\)

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 Feb 17, 2019

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