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[Solved] Functions and equations together

 
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June 15, 2021 11:13 am
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(@armor)
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Given     f(x)=x-\frac{1}{x},   solve the equation f[f(x)]=x.

 

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June 15, 2021 11:14 am
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small f[f(x)]=x-\frac{1}{x}-\frac{1}{x-\frac{1}{x}}=\frac{x^4-3x^2+1}{x^3-x},

thus  \small f[f(x)]=x\Leftrightarrow \frac{x^4-3x^2+1}{x^3-x}=x

\small \Rightarrow x^2=\frac{1}{2}\Rightarrow x=\pm \frac{\sqrt{2}}{2}.

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