Funções

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Problema

  MOSCOU-2000

Seja

f(x)=x^2 +12x +30.

Resolva a equação

f(f(f(f(f(x))))) =0.

Solução

(I)

    \begin{align*} f(x) & =  x^2 +12x +30\\ & = x^2 +12x+36-6\\ & = (x+6)^2-6 \end{align*}

(II)

    \begin{align*} f(\textcolor{blue}{f(x)}) & = (\textcolor{blue}{f(x)}+6)^2-6\\ & = [(x+6)^2-6+6]^2-6\\ & = (x+6)^4-6\\ \end{align*}

(III)

    \begin{align*} f(f(\textcolor{blue}{f(x)})) & = (\textcolor{blue}{f(x)}+6)^4-6\\ & = [(x+6)^2-6+6]^4-6\\ & = (x+6)^8-6\\ \end{align*}

(IV)

    \begin{align*} f(f(f(\textcolor{blue}{f(x)}))) & = (\textcolor{blue}{f(x)}+6)^8-6\\ & = [(x+6)^2-6+6]^8-6\\ & = (x+6)^{16} -6\\ \end{align*}

(V)

    \begin{align*} f(f(f(f(\textcolor{blue}{f(x)})))) & = (\textcolor{blue}{f(x)}+6)^{16} -6\\  = [(x+6)^{2}& -6+6]^{16}-6\\  = (x+6)^{32}&-6\\ \end{align*}

(VI)

    \begin{gather*} f(f(f(f(f(x)))))  = 0 \\ (x+6)^{32}-6  =0 \\ (x+6)^{32}=6  \\ x+6=\pm \sqrt[32]{6}\\ x = - 6 \pm \sqrt[32]{6} \end{gather*}

Assim

    \[ S=\{ - 6 \pm \sqrt[32]{6}\}.  \]

😉

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